group representations in banach spaces and banach...

Group representations inBanach spaces and Banach lattices

Proefschrift

ter verkrijging vande graad van Doctor aan de Universiteit Leiden,

op gezag van de Rector Magnificusprof. mr. P.F. van der Heijden,

volgens besluit van het College voor Promotieste verdedigen op woensdag 18 april 2012

klokke 16.15 uur

door

Marten Rogier Wortel

geboren te Delftin 1981

Samenstelling van de promotiecommissie:

promotor: Prof. dr. S.M. Verduyn Lunel

copromotores: Dr. M.F.E. de JeuProf. dr. B. de Pagter (Technische UniversiteitDelft)

Overige leden: Dr. O.W. van GaansProf. dr. N.P. Landsman (Radboud UniversiteitNijmegen)Prof. dr. P. StevenhagenProf. dr. A.W. Wickstead (Queen’s UniversityBelfast)

Group representations in

Banach spaces and Banach lattices

Contents

1 Introduction 7

2 Crossed products of Banach algebras 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.1 Group representations . . . . . . . . . . . . . . . . . . . . . . 272.2.2 Algebra representations . . . . . . . . . . . . . . . . . . . . . 292.2.3 Banach algebra dynamical systems and covariant representations 292.2.4 Cc(G,X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.5 Vector valued integration . . . . . . . . . . . . . . . . . . . . 322.2.6 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3 Construction and basic properties . . . . . . . . . . . . . . . . . . . . 342.4 Approximate identities . . . . . . . . . . . . . . . . . . . . . . . . . . 392.5 Representations: from (A,G, α) to (Aoα G)R . . . . . . . . . . . . . 452.6 Centralizer algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.7 Representations: from (Aoα G)R to (A,G, α) . . . . . . . . . . . . . 592.8 Representations: general correspondence . . . . . . . . . . . . . . . . 662.9 Representations: special correspondences . . . . . . . . . . . . . . . . 70

2.9.1 General algebra and group . . . . . . . . . . . . . . . . . . . 702.9.2 Trivial algebra: group Banach algebras . . . . . . . . . . . . . 732.9.3 Trivial group: enveloping algebras . . . . . . . . . . . . . . . 75

3 Positive representations of finite groups 773.1 Introduction and overview . . . . . . . . . . . . . . . . . . . . . . . . 773.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.3 Irreducible and indecomposable

representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.4 Structure of positive representations . . . . . . . . . . . . . . . . . . 89

3.4.1 Description of Aut+(Rn) . . . . . . . . . . . . . . . . . . . . . 893.4.2 Description of the finite subgroups of Aut+(Rn) . . . . . . . 903.4.3 Positive finite dimensional Archimedean

representations . . . . . . . . . . . . . . . . . . . . . . . . . . 923.4.4 Linear equivalence and order equivalence . . . . . . . . . . . 95

5

6 CONTENTS

3.5 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.5.1 Definitions and basic properties . . . . . . . . . . . . . . . . . 963.5.2 Frobenius reciprocity . . . . . . . . . . . . . . . . . . . . . . . 993.5.3 Systems of imprimitivity . . . . . . . . . . . . . . . . . . . . . 100

4 Compact groups of positive operators 1034.1 Introduction and overview . . . . . . . . . . . . . . . . . . . . . . . . 1034.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.3 Groups of positive operators . . . . . . . . . . . . . . . . . . . . . . . 1114.4 Representations with compact image . . . . . . . . . . . . . . . . . . 1164.5 Representations in Banach sequence spaces . . . . . . . . . . . . . . 1174.6 Positive representations in C0(Ω) . . . . . . . . . . . . . . . . . . . . 121

Bibliography 127

Dankwoord 131

Samenvatting 133

Curriculum Vitae 137

Chapter 1

Introduction

Motivated by quantum mechanics, amongst others, where there are many exam-ples of group representation in Hilbert spaces, strongly continuous unitary repre-sentations of locally compact groups have been studied extensively. In particu-lar, the decomposition theory is now well developed, which will now be illustratedby an example. Consider the group [0, 2π) with addition modulo 2π, which isisomorphic to the circle group S1, and the Hilbert space L2([0, 2π)). We exam-ine the left regular representation ρ of the group [0, 2π) in L2([0, 2π)) defined by(ρsf)(x) := f(x − s), for f ∈ L2([0, 2π)) and s, x ∈ [0, 2π). The collection of func-tions ein· := x 7→ einxn∈Z ⊂ L2([0, 2π)) forms an orthogonal basis of L2([0, 2π)),and the decomposition of a function f ∈ L2([0, 2π)) into this orthogonal basis is its

Fourier series f =∑n∈Z f(n)ein·, where

f(n) :=1

2π

∫ 2π

0

f(x)e−inx dx =1

2π

∫ 2π

0

f(x)einx dx (1.1)

denotes the n-th Fourier coefficient. Then, by the properties of the Fourier series,we obtain for s ∈ [0, 2π),

ρs

(∑n∈Z

f(n)ein·

)=∑n∈Z

e−insf(n)ein·. (1.2)

If we fix n ∈ Z, then the one dimensional subspace spanned by ein· is invariant underρ, and on this subspace the operator ρs is just a pointwise multiplication by e−ins.In particular, the restriction of ρ to this subspace is an irreducible representation.Hence ρ is an orthogonal direct sum over n ∈ Z of irreducible representations. Thisexample is a special case of the unitary representation theory of compact groups,which states that for any strongly continuous unitary representation of a compactgroup, the representation splits as an orthogonal direct sum of finite dimensionalirreducible representations.

For non-compact locally compact groups such a direct sum decomposition isnot always possible, but it is still possible to view the original representation as

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8 CHAPTER 1. INTRODUCTION

somehow “built up” from irreducible ones. The following example, which is similarto the above example, explains how this is done. Let G be any abelian locallycompact group with a Haar measure µ, i.e., a left invariant regular measure on Gwhich is finite on compact sets. Again we consider the left regular representation ρof G on L2(G,µ) defined by (ρsf)(r) := f(r − s), for s, r ∈ G. Consider the dualgroup Γ = Hom(G,S1). The dual group, equipped with the compact-open topology,is again a locally compact abelian group. An example is G = R with its naturaltopology, then Γ ∼= R with its natural topology. The Fourier transform f 7→ f fromL1(G,µ) to C0(Γ) defined by

f(γ) :=

∫G

f(r)γ(r) dµ(r), γ ∈ Γ

is a generalization of (1.1), and its restriction to L1(G,µ) ∩ L2(G,µ) maps iso-metrically into L2(Γ, λ), where λ is an appropriately chosen Haar measure on Γ,and it extends to an isometric isomorphism of Hilbert spaces between L2(G,µ) andL2(Γ, λ). So we may transport our representation ρ from L2(G,µ) to L2(Γ, λ), andthere we obtain, using the properties of the Fourier transform, for f ∈ L2(G,µ),s ∈ G and almost every γ ∈ Γ,

(ρsf)(γ) = γ(s)f(γ),

which is similar to (1.2); here the representation corresponds to a pointwise almosteverywhere multiplication by γ(s), some sort of “integral” of pointwise multiplica-tions, in particular, of irreducible representations. This can be formalized using thenotion of direct integrals of Hilbert spaces and direct integrals of representations,and using this notion, we can state the main theorem on decomposing strongly con-tinuous unitary representations of locally compact groups in terms of irreduciblerepresentations, cf. [50, Corollary 14.9.5].

Theorem 1.1. Let G be a separable locally compact group, H a separable Hilbertspace and ρ a strongly continuous unitary representation of G on H. Then H isisometrically isomorphic to a direct integral of Hilbert spaces, such that under thisisomorphism, the representation ρ corresponds to a direct integral of irreducible rep-resentations.

Unitary representations often arise naturally whenever a group acts on a set, e.g.,let X ⊂ C be the closed unit disc with Lebesgue measure, then S1 acts naturallyon X and a strongly continuous unitary representation ρ of S1 in L2(X) is definedby (ρsf)(x) := f(s−1x), for s ∈ S1, f ∈ L2(X) and x ∈ X. By the above suchrepresentations can be decomposed into irreducibles. However, the same formulayields a strongly continuous representation of S1 in the Banach spaces Lp(X), for1 ≤ p < ∞, and in C(X). Hence such representations on Banach spaces occurnaturally - what about these representations? Do we have a similar decompositiontheory as in the unitary case?

This thesis is a contribution to the theory of such representations. It consistsof two parts. The first part, Chapter 2, is about the crossed product. The second

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part, Chapters 3 and 4, is about positive representations in partially ordered vectorspaces. We will now discuss the first part.

Crossed products

When studying group representations, it is often useful to look at algebras. Anexample is the group algebra k[G] of a finite group G over a field k. This algebra hasthe property that there is a bijection between representations of G on k-vector spacesand algebra representations of k[G] on the same vector space, and so questions aboutgroup representations can be translated into questions about algebra representations.

In the theory of unitary representations, such an algebra also exists. Given alocally compact group G, this object is the group C∗-algebra C∗(G), a C∗-algebra forwhich the nondegenerate algebra representations in a Hilbert space are in bijectionwith the strongly continuous unitary representations of the group in that Hilbertspace. The C∗-algebra C∗(G) is a crucial tool in proving Theorem 1.1. Indeed, allthe hard work lies in proving a similar fact about representations of C∗-algebras onHilbert spaces, and then the result about unitary representations follows immediatelyby applying this to C∗(G).

In view of the above it is desirable, given a locally compact group G, to havea Banach algebra such that some of its algebra representations in certain Banachspaces are in bijection with some of the strongly continuous group representations ofG in the same class of Banach spaces. The reason for not considering all representa-tions is the following. In the Hilbert space case, there is up to isomorphism only oneseparable infinite dimensional Hilbert space. However, there is a great diversity ofinfinite dimensional separable Banach spaces, and to consider all representations inall these spaces seems a daunting task. It would be much better if the above Banachalgebra can be specialized to specific situations. E.g, if one is interested in stronglycontinuous isometric group representations in Lp-spaces for some p ≥ 1, then onewould want a Banach algebra such that its algebra representations in Lp-spaces arein bijection with these group representations. Or, if one is interested in uniformlybounded representations in spaces of continuous functions, then one would want adifferent Banach algebra with a bijection concerning these representations in thesespaces. In other words, the construction of the group C∗-algebra needs to be gener-alized such that it can be adapted to whatever representations one is interested in,instead of only the unitary group representations.

The group C∗-algebra is a special case of a more general object called a crossedproduct C∗-algebra, which is not only useful in translating unitary group represen-tations to algebra representations, but also has applications concerning induced rep-resentations of subgroups. Hence we will generalize the crossed product C∗-algebra,which we will now briefly discuss.

A C∗-dynamical system is a triple (A,G, α), where A is a C∗-algebra, G is alocally compact group and α : G → Aut(A) is a strongly continuous action of Gon A. This can be viewed as a generalization of a locally compact group, sinceif A = C, the action α has to be trivial and G is the only nontrivial object. Acovariant representation of (A,G, α) is a pair (π, U), where π is a representation of


A in a Hilbert space, and U is a strongly continuous unitary representation of G inthe same Hilbert space, satisfying the covariance relation

Usπ(a)U−1s = π(αs(a)),

for all s ∈ G and a ∈ A. Again, if A = C, π is trivial and hence the covariantrelation is always satisfied, and the group representation is the only nontrivial ob-ject. Hence studying covariant representations of (C, G, α) is the same as studyingstrongly continuous unitary representations of G. The crossed product C∗-algebrais a C∗-algebra AoαG with the property that the class of covariant representationsof (A,G, α) is in bijection with the class of nondegenerate representations of AoαGin Hilbert spaces. If A = C, we recover the group C∗-algebra C∗(G).

The above can be generalized to the Banach algebra case as follows. A Ba-nach algebra dynamical system is a triple (A,G, α) with the same properties asa C∗-dynamical system, except that A is only assumed to be a Banach algebra.Covariant representations are generalized by allowing the representations to be inBanach spaces instead of Hilbert spaces. The main difference is in the class of co-variant representations being considered; in the Hilbert space case, this class equalsall covariant representations in Hilbert spaces. Since, as explained earlier, we wantto vary the class of covariant representations being considered, this class is an addi-tional variable, which will be called R, going into the crossed product construction.It turns out that one cannot consider all classes R. There has to be some uniformbound on the norm of the algebra representations, and the norm of the group rep-resentations has to be uniformly bounded by some fixed function ν : G → [0,∞)which is bounded on compact sets, i.e., ‖Ur‖ ≤ ν(r) for all (π, U) ∈ R. Note thatthese requirements are automatically satisfied in the C∗-algebra case, as C∗-algebrarepresentations in Hilbert spaces are contractive and unitary representations areisometric. Given such a class R, one also needs to define the class of R-continuouscovariant representations (Definition 2.5.1), which is in general a larger class thanR. In the C∗-algebra case, these classes coincide.

A technical obstacle that we encountered while generalizing the crossed productC∗-algebra is as follows. In the C∗-algebra case, it is at some point necessary toextend a bounded nondegenerate representation of a C∗-algebra to its multiplieralgebra, which can be done easily using C∗-theory. The Banach algebra analogueof this is, given a nondegenerate bounded representation of a Banach algebra, toobtain an extension of this representation to centralizer algebras of the originalBanach algebra. It turns out that this can be done in a satisfactory manner, whichwas worked out in [9]. After overcoming this technical obstacle, and incorporatingthe new features concerning the R-continuous covariant representations, it turns outthat the crossed product C∗-algebra can indeed be generalized, cf. Theorem 2.8.1, themain result of Chapter 2. It states that, given a Banach algebra dynamical systemwith a mild condition on A (it has to have a bounded approximate left identity), aclassR as above and a class X of Banach spaces, there is a Banach algebra (AoαG)R

such that its bounded nondegenerate algebra representations in spaces from X arein bijection with the class of R-continuous covariant representations of (A,G, α) inspaces from X .

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We have specialized the above result to certain natural classes of group repre-sentations in Banach spaces, cf. Section 2.9, and this allows us to study such grouprepresentations by studying the representations of the specific Banach algebras thusobtained instead. Further investigation of these special Banach algebras is neededto optimally exploit this result, but at least the problem has become more tractablenow from a functional analytical point of view, since a Banach algebra has morefunctional analytic structure than a group. Given the success of the archetypicaltransition from the group to the group C∗-algebra in the case of unitary group rep-resentations, the availability of such a transition, “tuned to the situation at hand”,can be considered as a step forward towards a decomposition theorem for otherclasses of group representations than the unitary ones.

Another type of group representations one would like to understand better arethe positive group representations, which we will discuss in the next section. Forsuch representations, the appropriate Banach algebra of crossed product type willhave to be a so-called Banach lattice algebra, and the construction of the crossedproduct needs further modification. We leave this for further research, noting thatthe results and techniques in Chapter 2 provide a concrete model to start from.

For unitary representation of compact groups, however, it is already possibleto obtain the existence of a decomposition into irreducibles without the use of thegroup C∗-algebra. One might hope that a similar phenomenon occurs for positiverepresentations of compact groups. As is shown in Chapters 3 and 4, which constitutethe second part of this thesis, for certain spaces this is indeed the case.

We will now specialize our discusssion from general Banach space representationsto positive representations.

Positive representations

We return again to our motivating example of the representation ρ of S1 in Lp(X)(1 ≤ p <∞) and C(X), where X ⊂ C is the closed unit disc, defined by

(ρsf)(x) := f(s−1x),

for s ∈ S1 and x ∈ X. It is clear that, for s ∈ S1, the maps ρs are positive, i.e., theymap positive functions to positive functions. This positivity of the operators ρs isthe context for Chapter 3 and Chapter 4.

The natural partial order on functions spaces such as Lp(X), i.e., f ≥ g if andonly if f(x) ≥ g(x) for almost every x, can be studied by the following abstraction.A partially ordered vector space is a real vector space equipped with a partial orderthat is compatible with the vector space structure, i.e., if the vectors x and y arepositive, then x + y is positive, and if λ is a positive scalar, then λx is positive. ARiesz space is a partially ordered vector space where each pair of vectors x and yhas a supremum x ∨ y and an infimum x ∧ y. In a Riesz space one can define anabsolute value as in function spaces, by |x| := x ∨ (−x), and x and y are calleddisjoint if |x| ∧ |y| = 0. If L is a subset of a Riesz space, then Ld denotes thedisjoint complement of A, i.e., all vectors that are disjoint from all vectors from


A. A Banach lattice is a Riesz space, and a Banach space, such that the norm iscompatible with the order structure, i.e., if x and y are vectors such that |x| ≤ |y|,then ‖x‖ ≤ ‖y‖. Many function spaces considered in analysis, such as Lp-spaces andspaces of continuous functions, are Banach lattices.

By the above it makes sense to study positive representations in Banach lat-tices, as they appear naturally: whenever there is a group acting on some set, moreoften than not there is an induced positive representation in a Banach lattice offunctions defined on that set. Very little is known about positive representations.The natural question in this case, similar to the unitary case, is whether a positiverepresentation in a Banach lattice can be decomposed into order indecomposablesubrepresentations. This will be the main theme in our investigations of positiverepresentations.

In Chapter 3 we consider the simplest case: the finite dimensional case. Sincevector space topologies are not interesting in the finite dimensional setting, we lookat the more general setting of Riesz spaces, instead of Banach lattices. We are inter-ested in decompositions, and a natural question is whether an order indecomposablepositive representation of a finite group is finite dimensional. With an order inde-composable positive representation we mean a positive representations such that theRiesz space cannot be written as the order direct sum of two subspaces that are bothinvariant under the representation. An order direct sum of a Riesz space E means adirect sum E = L⊕M , such that whenever x = y+z ∈ E is positive, with y ∈ L andz ∈M , then y and z are positive. In this case L and M are automatically so-calledprojection bands which are each other’s disjoint complement, so E = L⊕Ld. This issimilar to the Hilbert space case, where a closed linear subspace L of a Hilbert spaceH induces an orthogonal decomposition H = L⊕L⊥. Moreover, if a projection bandis invariant under a positive representation of a group, its disjoint complement isalso invariant, which is similar to the Hilbert space case where the orthogonal ana-logue holds for a unitary representation of a group. This easily implies that orderindecomposability of a positive representation in a Riesz space is equivalent withits projection band irreducibility, i.e., that it does not possess a nontrivial properinvariant projection band. Again, this is similar to the case of unitary representa-tions, where indecomposability is equivalent with irreducibility, i.e., with the absenseof nontrivial proper invariant closed subspaces. Hence the above question can bereformulated as: is a projection band irreducible positive representation of a finitegroup finite dimensional?

The corresponding question in the unitary case is trivially true. Indeed, let ρ bean irreducible unitary representation of a finite group in a Hilbert space H. Take anonzero vector x ∈ H and consider the subspace generated by its orbit under ρ. Thissubspace is finite dimensional and hence closed, and by construction ρ-invariant, soit must equal the whole space H, hence H is finite dimensional. Unfortunately thisproof breaks down in the ordered setting, as bands, in particular projection bands,are generally infinite dimensional, e.g., in Lp([0, 1]), for any p, all nontrivial bandsare infinite dimensional.

In the ordered case, the above question has a negative answer. Indeed, therepresentation of the trivial group in C([0, 1]) is projection band irreducible. In this

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example the cause lies with the Riesz space, one might say, as C([0, 1]) does nothave any proper nontrivial projection bands at all: it is far from being what is calledDedekind complete. All Lp-spaces, on the other hand, are Dedekind complete.

If we assume that the Riesz space is Dedekind complete, then the situationimproves. In this case we managed to show, with an unusual proof based on inductionon the order of the group, that if a positive representation of a finite group in aDedekind complete Riesz space is projection band irreducible, then the Riesz spaceis finite dimensional, cf. Theorem 3.3.14. In this theorem we actually prove a littlebit more, but this is the most important consequence.

Having obtained this result, we then looked at the positive representations offinite groups in finite dimensional Archimedean Riesz space. These spaces are orderisomorphic to Rn for some n ∈ N with pointwise ordering. We first study the auto-morphism group of Rn, i.e., the group of positive matrices with positive inverses, andshow that it equals the semidirect product of the subgroup of diagonal matrices withstrictly positive entries on the diagonal, and the subgroup of permutation matrices.Using some basic group cohomology methods, it turns out that every finite groupof positive matrices equals a group of permutation matrices conjugated by a singlediagonal matrix with strictly positive diagonal elements, and from this it followseasily that every positive representation equals a permutation representation conju-gated by such a diagonal matrix. From this we obtain a nice characterization of theorder dual of a finite group, i.e., the space of order equivalence classes of irreduciblepositive representations, in terms of group actions on finite sets, cf. Theorem 3.4.10,which is as follows.

Theorem 1.2. Let G be a finite group. If H ⊂ G is a subgroup, let EH be the|G : H|-dimensional vector space of real-valued functions on G/H, equipped withthe pointwise ordering. Let πH be the canonical positive representation of G in EH ,corresponding to the action of G on G/H. Then, whenever H1 and H2 are conjugate,πH1 and πH2 are order equivalent, and the map

[H] 7→ [πH ]

is a bijection between the conjugacy classes of subgroups of G and the order equiva-lence classes of irreducible positive representations of G in nonzero finite dimensionalArchimedean Riesz spaces.

Additionally, we obtain a unique decomposition of positive representations of finitegroups in finite dimensional Archimedean Riesz spaces into band irreducibles.

We also show that characters do not, in general, determine representations, inthe sense that there even exist band irreducible positive finite dimensional represen-tations of finite groups, having the same character, which are not order isomorphic.Finally, we look at induction in the ordered setting, the categorical aspects of whichare largely the same as in the nonordered setting, but for which the multiplicityversion of Frobenius reciprocity turns out not to hold.

In Chapter 4 we take the above results to the next level: that of compact groups.As the image of a strongly continuous representation of a compact group in a Ba-


nach lattice is a group of positive operators which is compact in the strong operatortopology, such compact groups of positive operators are investigated. We assumethat these groups are contained in the product, which again is a semidirect prod-uct, of the group of central lattice automorphisms and the group of isometric latticeautomorphisms. This is motivated by the above results on representations in Rnequipped with any of the p-norms, where the isometric lattice automorphisms arethe permutation matrices and the central lattice automorphisms are the diagonalmatrices with strictly positive elements on the diagonal, and so the whole automor-phism group of Rn equals this semidirect product and hence every subgroup satisfiesthis assumption. This characterization of the automorphism group is also satisfiedin many natural sequence spaces and spaces of continuous functions. However, notevery Banach lattice has such a nice characterization of the automorphism group.

Under the additional technical Assumption 4.3.3 which is satisfied in many nat-ural sequence spaces and spaces of continuous functions, we are able to obtain, as inthe finite dimensional case, that such a compact group equals a group of isometriclattice automorphisms conjugated by a single central lattice automorphism. This isespecially useful in the aforementioned spaces, as we have a nice description availableof both the isometric lattice automorphisms and the central lattice automorphisms.Again this leads to a similar description of strongly continuous positive represen-tations of compact groups with range in this product: it is a strongly continuousisometric positive representation conjugated by a single central lattice automor-phism. Since everything depends only on the compactness in the strong operatortopology of the image of the representation, we have the same result for arbitraryrepresentations of arbitrary groups with compact image. Applying these results tothe sequence space case, we obtain the following ordered analogue of the aforemen-tioned theorem on the decomposition of strongly continuous unitary representationsof compact groups, which is as follows, cf. Theorem 4.5.7.

Theorem 1.3. Let E be a normalized symmetric Banach sequence space, let G bea group and let ρ be a positive representation of G in E. Then E splits into bandirreducibles, in the sense that there exists an α with 1 ≤ α ≤ ∞ such that the setof invariant and band irreducible bands Bn1≤n≤α (if α < ∞) or Bn1≤n<∞ (ifα =∞) satisfies

x =

α∑n=1

Pnx ∀x ∈ E, (1.3)

where Pn : E → Bn denotes the band projection, and the series is unconditionallyorder convergent, hence, in the case that E has order continuous norm, uncondi-tionally convergent.

Moreover, if ρ has compact image and E has order continuous norm or E = `∞,then every invariant and band irreducible band is finite dimensional, and so α =∞.

Examples of normalized symmetric Banach sequence spaces with order continuousnorm are the classical sequence spaces c0 and `p for 1 ≤ p <∞.

In general, one cannot expect a direct sum decomposition into band irreduciblesas in the above theorem for positive representations of compact groups in arbitrary

15

Banach lattices. An example is the representation of the trivial group in Lp([0, 1]),1 ≤ p ≤ ∞, where there are no nonzero band irreducible subrepresentations atall. In order to obtain some kind of decomposition in other spaces, some new ideasare needed. A result in this direction is [21], in which composition series of orderedstructures are examined. Another result is [23], where positive representations of Lp-spaces associated with Polish transformation groups are considered. In that paper itis shown that for such representations, a decomposition into band irreducibles exists,in terms of Banach bundles, which is at least in spirit close to the direct integral ofHilbert spaces used in Theorem 1.1.

It is clear that there is still a lot of work to be done concerning decompositiontheorems for positive representations, but the first steps have now been taken.

Chapter 2

Crossed products of Banachalgebras

This chapter is to appear in Dissertationes Mathematicae as: M. de Jeu, S. Dirk-sen and M. Wortel, “Crossed products of Banach algebras. I.”. It is available asarXiv:1104.5151.

Abstract. We construct a crossed product Banach algebra from a Banach algebradynamical system (A,G, α) and a given uniformly bounded class R of continuouscovariant Banach space representations of that system. If A has a bounded leftapproximate identity, and R consists of non-degenerate continuous covariant rep-resentations only, then the non-degenerate bounded representations of the crossedproduct are in bijection with the non-degenerate R-continuous covariant represen-tations of the system. This bijection, which is the main result of the paper, is alsoestablished for involutive Banach algebra dynamical systems and then yields thewell-known representation theoretical correspondence for the crossed product C∗-algebra as commonly associated with a C∗-algebra dynamical system as a specialcase. Taking the algebra A to be the base field, the crossed product constructionprovides, for a given non-empty class of Banach spaces, a Banach algebra with arelatively simple structure and with the property that its non-degenerate contrac-tive representations in the spaces from that class are in bijection with the isometricstrongly continuous representations of G in those spaces. This generalizes the no-tion of a group C∗-algebra, and may likewise be used to translate issues concerninggroup representations in a class of Banach spaces to the context of a Banach algebra,simpler than L1(G), where more functional analytic structure is present.

2.1 Introduction

The theory of crossed products of C∗-algebras started with the papers by Turu-maru [49] from 1958 and Zeller-Meier from 1968 [54]. Since then the theory has

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18 CHAPTER 2. CROSSED PRODUCTS OF BANACH ALGEBRAS

been extended extensively, as is attested by the material in Pedersen’s classic [35]and, more recently, in Williams’ monograph [51]. Starting with a C∗-dynamicalsystem (A,G, α), where A is a C∗-algebra, G is a locally compact group, and αa strongly continuous action of G on A as involutive automorphisms, the crossedproduct construction yields a C∗-algebra A oα G which is built from these data.Thus the crossed product construction provides a means to construct examples ofC∗-algebras from, in a sense, more elementary ingredients. One of the basic factsfor a crossed product C∗-algebra Aoα G is that the non-degenerate involutive rep-resentations of this algebra on Hilbert spaces are in one-to-one correspondence withthe non-degenerate involutive continuous covariant representations of (A,G, α), i.e.,with the pairs (π, U), where π is a non-degenerate involutive representation of A ona Hilbert space, and U is a unitary strongly continuous representations of G on thesame space, such that the covariance condition π(αs(a)) = Usπ(a)U−1

s is satisfied,for a ∈ A, and s ∈ G.

This paper contains the basics for the natural generalization of this constructionto the general Banach algebra setting. Starting with a Banach algebra dynamicalsystem (A,G, α), where A is a Banach algebra, G is a locally compact group, and α astrongly continuous action of G on A as not necessarily isometric automorphisms, wewant to build a Banach algebra of crossed product type from these data. Moreover,we want the outcome to be such that (suitable) non-degenerate continuous covariantrepresentations of (A,G, α) are in bijection with (suitable) non-degenerate boundedrepresentations of this crossed product Banach algebra. Later in this introduction,more will be said about how to construct such an algebra, and how the constructioncan be tuned to accommodate a class R of non-degenerate continuous covariantrepresentations relevant for the case at hand. It will then also become clear whatbeing “suitable” means in this context. For the moment, let us oversimplify a bitand, neglecting the precise hypotheses, state that such an algebra can indeed beconstructed. The precise statement is Theorem 2.8.1, which we will discuss below.

Before continuing the discussion of crossed products of Banach algebra as such,however, let us mention our motivation to start investigating these objects, andsketch perspectives for possible future applications of our results. Firstly, just as inthe case of a crossed product C∗-algebra, it simply seems natural as such to have ameans available to construct Banach algebras from more elementary building blocks.Secondly, there are possible applications of these algebras in the theory of Banachrepresentations of locally compact groups. We presently see two of these, which wewill now discuss.

Starting with the first one, we recall that, as a special case of the correspon-dence for crossed product C∗-algebras mentioned above, the unitary strongly con-tinuous representations of a locally compact group G in Hilbert spaces are in bi-jection with the non-degenerate involutive representations of the group C∗-algebraC∗(G) = C otriv G in Hilbert spaces. It is by this fact that questions concern-ing, e.g., the existence of sufficiently many irreducible unitary strongly continuousrepresentations of G to separate its points, and, notably, the decomposition of anarbitrary unitary strongly continuous representation of G into irreducible ones, canbe translated to C∗-algebras and solved in that context [10]. For Banach space

2.1. INTRODUCTION 19

representations of G, the theory is considerably less well developed. To our knowl-edge, the only available general decomposition theorem, comparable to those in aunitary context, is Shiga’s [46], stating that a strongly continuous representation ofa compact group in an arbitrary Banach space decomposes in a Peter-Weyl–fashion,analogous to that for a unitary strongly continuous representation in a Hilbert space.With the results of the present paper, it is possible to construct Banach algebraswhich, just as the group C∗-algebra, are “tuned” to the situation. Our main resultsin this direction are Theorem 2.9.7 and Theorem 2.9.8. The latter, for example,yields, for any non-empty class X of Banach spaces, a Banach algebra BX (G), suchthat the non-degenerate contractive representations of BX (G) in spaces from X arein bijection with the isometric strongly continuous representations of G in thesespaces. This algebra BX (G) could be called the group Banach algebra of G associ-ated with X , and, as will become clear in Section 2.9.2, only the isometric stronglycontinuous representations of G in the spaces from X are used in its construction.The analogy with the group C∗-algebra C∗(G), which is in fact a special case, isclear. Just as is known to be the case with C∗(G), one may hope that, for certainclasses X of sufficiently well-behaved spaces, the study of BX (G) will shed light onthe theory of isometric strongly continuous representations of G in the spaces fromX . For comparison, we recall the well-known fact [20, Assertion VI.1.32], [24, Propo-sition 2.1] that the non-degenerate bounded representations of L1(G) in a Banachspace are in bijection with the uniformly bounded strongly continuous representa-tions of G in that Banach space. So, certainly, there is already a Banach algebraavailable to translate questions concerning group representations to, but the point isthat it is very complicated, simply because L1(G) apparently carries the informationof all such representations of G in all Banach spaces. One may hope that, for certainchoices of X , an algebra such as BX (G), the construction of which uses no more datathan evidently necessary, has a sufficiently simpler structure than L1(G) to admitthe development of a reasonable representation theory, and hence for the isometricstrongly continuous representations of G in these spaces, thus paralleling the casewhere X consists of all Hilbert spaces and BX (G) = C∗(G). Aside, let us mentionthat L1(G) is, in fact, isometrically isomorphic to a crossed product (FotrivG)R as inthe present paper, if one chooses the class R—to be discussed below—appropriately.In that case, it is possible to infer the aforementioned bijection between the non-degenerate bounded representations of L1(G) and the uniformly bounded stronglycontinuous representations of G from Theorem 2.8.1, due to the fact that these rep-resentations of G can then be seen to correspond to the R-continuous—also to bediscussed below—representations of (F, G, triv). In view of the further increase inlength of the present paper that would be a consequence of the inclusion of theseand further related results, we have decided to postpone these to the sequel [22],including only some preparations for this at the end of Section 2.9.1.

The second possible application in group representation theory lies in the relationbetween imprimitivity theorems and Morita equivalence. Whereas the constructionof the group Banach algebras BX (G) and establishing their basic properties could bedone in a paper quite a bit shorter than the present one, the general crossed prod-uct construction and ensuing results are indeed needed for this second perspective.


Starting with the involutive context, we recall that Mackey’s now classical result [30]asserts that a unitary strongly continuous representation U of a separable locallycompact group G is unitarily equivalent to an induced unitary strongly continuousrepresentation of a closed subgroup H, precisely when there exists a system of im-primitivity (G/H,U, P ) based on the G-space G/H. The separability condition of Gis actually not necessary, as shown by Loomis [27] and Blattner [5], and for generalG Mackey’s imprimitivity theorem can be reformulated as ([40, Theorem 7.18]): Uis unitarily equivalent to such an induced representation precisely when there ex-ists a non-degenerate involutive representation π of C0(G/H) in the same Hilbertspace, such that (π, U) is a covariant representation of the C∗-dynamical system(C0(G/H), G, lt), where G acts as left translations on C0(G/H). Using the standardcorrespondence for crossed products of C∗-algebras, one thus sees that, up to unitaryequivalence, such U are precisely the unitary parts of the non-degenerate involutivecontinuous covariant representations of the C∗-dynamical system (C0(G/H), G, lt)corresponding to the non-degenerate involutive representations of the crossed prod-uct C∗-algebra C0(G/H)oltG. Rieffel’s theory of induction for C∗-algebras [37], [40]and Morita-equivalence [38], [41] allows us to follow another approach to Mackey’stheorem, as was in fact done in [40], by proving that C0(G/H)oltG and C∗(H) are(strongly) Morita equivalent as a starting point. This implies that these C∗-algebrashave equivalent categories of non-degenerate involutive representations, and workingout this correspondence then yields Mackey’s imprimitivity theorem. For more de-tailed information we refer to [40], [38], and [41], as well as (also including significantfurther developments) to [15], [31], [36], [12], [51] and [14], the latter also for Banach∗-algebras and Banach ∗-algebraic bundles.

The Morita theorems in a purely algebraic context are actually more symmet-ric than the analogous ones in Rieffel’s work. We formulate part of the results foralgebras over a field k (cf. [13, Theorem 12.12]): If A and B are unital k-algebras,then the categories of left A-modules and left B-modules are k-linearly equivalentprecisely when there exist bimodules APB and BQA, such that P ⊗B Q ' A asA-A-bimodules, and Q ⊗A P ' B as B-B-bimodules. From the existence of suchbimodules it follows easily that the categories are equivalent, since equivalence aremanifestly given by M 7→ Q⊗AM , for a left A-module M , and by N 7→ P ⊗BN , ora left B-module N . The non-trivial statement is that the converse is equally true. InRieffel’s analytical context, the role of the bimodules P and Q for C∗-algebras A andB is taken over by so-called imprimitivity bimodules, sometimes also called equiva-lence bimodules. These are A-B-Hilbert C∗-modules ([36, Definition 3.1]), and theexistence of such imprimitivity bimodules (actually, exploiting duality, only one isneeded, see [36, p. 49]) implies that the categories of non-degenerate involutive rep-resentations of these C∗-algebras are equivalent [36, Theorem 3.29]. In contrast withthe algebraic context, the converse is not generally true (see [36, Remark 3.15 andHooptedoodle 3.30]). This has led to the distinction between strong Morita equiva-lence (in the sense of existing imprimitivity bimodules) and weak Morita equivalence(in the sense of equivalent categories of non-degenerate involutive representations)of C∗-algebras. The work of Blecher [7], generalizing earlier results of Beer [4],shows how to remedy this: if one enlarges the categories, taking them to consist


of all left A-operator modules as objects and completely bounded A-linear maps asmorphisms, and similarly for B, then symmetry is restored as in the algebraic case:the equivalence of these larger categories is then equivalent with the existence of animprimitivity bimodule, i.e., with strong Morita equivalence of the C∗-algebras inthe sense of Rieffel. As a further step, strong Morita equivalence was developed foroperator algebras (i.e., norm-closed subalgebras of B(H), for some Hilbert space H)by Blecher, Muhly and Paulsen in [8]. Restoring symmetry again, Blecher proved in[6] that, for operator algebras with a contractive approximate identity, strong Moritaequivalence is equivalent to their categories of operator modules being equivalent viacompletely contractive functors.

A part of the well-developed theory in a Hilbert space context as mentioned abovehas a parallel for Banach algebras and representations in Banach spaces, but, as faras we are aware, the body of knowledge is much smaller than for Hilbert spaces.1 In-duction of representations of locally compact groups and Banach algebras in Banachspaces has been investigated by Rieffel in [39], from the categorical viewpoint that,as a functor, induction is, or ought to be, an adjoint of the restriction functor. In[18], Grønbæk studies Morita equivalence for Banach algebras in a context of Banachspace representations, and a Morita-type theorem [16, Corollary 3.4] is establishedfor Banach algebras with bounded two-sided approximate identities: such Banachalgebras A and B have equivalent categories of non-degenerate left Banach modulesprecisely when there exist non-degenerate Banach bimodules APB and BQA, suchthat P ⊗BQ ' A as A-A-bimodules, and Q⊗AP ' B as B-B-bimodules. In subse-quent work [17], this result is generalized to self-induced Banach algebras, and thisgeneralization yields an imprimitivity theorem [18, Theorem IV.9] in a form quitesimilar to Mackey’s theorem as formulated by Rieffel [40, Theorem 7.18] (i.e., witha C0(G/H)-action instead of a projection valued measure), with a continuity con-dition on the action of C0(G/H). The approach of this imprimitivity theorem, viaMorita equivalence of Banach algebras, is therefore analogous to Rieffel’s work, andhere again algebras which are called crossed products make their appearance [18,Definition IV.1]. Given the results in the present paper, it is natural to ask whetherthis imprimitivity theorem (or a variation of it) can also be derived from a surmisedMorita equivalence of the crossed product Banach algebra (C0(G/H) olt G)R anda group Banach algebra BX (H) as in the present paper (for suitable R and X ),and what the relation is between the algebras in [18, Definition IV.1], also calledcrossed products, and the crossed product Banach algebras in the present paper.We expect to investigate this in the future, also taking the work of De Pagter andRicker [34] into account. In that paper, it is shown that, for certain bounded Banachspace representations (including all bounded representations in reflexive spaces2) ofC(K), where K is a compact Hausdorff space, there is always an underlying pro-jection valued measure. In such cases, if G/H is compact (and it is perhaps notoverly optimistic to expect that the results in [34] can be generalized to the locally

1As an illustration: as far as we know, for groups, [29] is currently the only available book onBanach space representations.

2In fact: including all bounded representations in spaces not containing a copy of c0, see [34,Corollary 2.16].


compact case, so as to include representations of C0(G/H) for non-compact G/H),an imprimitivity theorem for Banach space representations of groups can be derivedin Mackey’s original form in terms of systems of imprimitivity. If all this comesto be, then this would be a satisfactory parallel—for suitable Banach spaces—withthe Hilbert space context, both in the spirit of Rieffel’s strong Morita equivalenceof C0(G/H) olt G and C∗(H) as a means to obtain an imprimitivity theorem, andof Mackey’s systems of imprimitivity as a means to formulate such a theorem. Wehope to be able to report on this in due time.

We will now outline the mathematical structure of the paper. Although thecrossed product of a general Banach algebra is more involved than its C∗-algebracounterpart, the reader may still notice the evident influence of [51] on the presentpaper. We start by explaining how to construct the crossed product. Given a Banachalgebra dynamical system (A,G, α) (Definition 2.2.10), and a non-empty class R ofcontinuous covariant representations (Definition 2.2.12), we want to introduce analgebra seminorm σR on the twisted convolution algebra Cc(G,A) by defining

σR(f) = sup(π,U)∈R

∥∥∥∥∫G

π(f(s))Us ds

∥∥∥∥ (f ∈ Cc(G,A)).

For a C∗-dynamical system, if one lets R consist of all pairs (π, U) where π is involu-tive and non-degenerate, and U is unitary and strongly continuous, this supremumis evidently finite, and σR is even a norm. For a general Banach algebra dynamicalsystem, neither need be the case. This therefore leads us, first of all, to introducethe notion of a uniformly bounded (Definition 2.3.1) class of covariant representa-tions, in order to ensure the finiteness of σR. The resulting crossed product Banachalgebra (A oα G)R is then, by definition, the completion of Cc(G,A)/ker (σR) inthe algebra norm induced by σR on this quotient. Thus, as a second difference withthe construction of the crossed product C∗-algebra associated with a C∗-dynamicalsystem, a non-trivial quotient map is inherent in the construction.

While the construction is thus easily enough explained, the representation the-ory, to which we now turn, is more involved. Suppose that (π, U) is a continu-ous covariant representation of (A,G, α), and that there exists C ≥ 0, such that∥∥∫Gπ(f(s))Us ds

∥∥ ≤ CσR(f), for all f ∈ Cc(G,A). In that case, we say that (π, U)is R-continuous, and it is clear that there is an associated bounded representationof (Aoα G)R, denoted by (π oU)R. Certainly all elements of R are R-continuous,yielding even contractive representations of (A oα G)R, but, as it turns out, theremay be more. Likewise, (A oα G)R may have non-contractive bounded represen-tations. This contrasts the analogous involutive context for the crossed productC∗-algebra associated with a C∗-dynamical system. The natural question is, then,what the precise relation is between the R-continuous covariant representations of(A,G, α) and the bounded representations of (Aoα G)R. The answer turns out tobe quite simple: if A has a bounded left approximate identity, and if R consists ofnon-degenerate (Definition 2.2.12) continuous covariant representations only, thenthe map (π, U) 7→ (πoU)R is a bijection between the non-degenerate R-continuous


covariant representations of (A,G, α) and the non-degenerate bounded representa-tions of (Aoα G)R. This is the main content of Theorem 2.8.1.

Establishing this, however, is less simple. The first main step to be taken is toconstruct any representations of the group and the algebra at all from a given (non-degenerate) bounded representation of (A oα G)R. In case of a crossed productC∗-algebra and involutive representations in Hilbert spaces, there is a convenientway to proceed [51]. One starts by viewing this crossed product as an ideal of itsdouble centralizer algebra. If the involutive representation T of the crossed productC∗-algebra is non-degenerate, then it can be extended to an involutive representationof the double centralizer algebra. Subsequently, it can be composed with existinghomomorphisms of group and algebra into this double centralizer algebra, thus yield-ing a pair (π, U) of representations. These can then be shown to have the desiredcontinuity, involutive and covariance properties and, moreover, the correspondingnon-degenerate involutive representation of the crossed product C∗-algebra turnsout to be T again. For Banach algebra dynamical systems we want to use a simi-lar circle of ideas, but here the situation is more involved. To start with, it is notnecessarily true that a Banach algebra A can be mapped injectively into its doublecentralizer algebra M(A), or that a non-degenerate representation of A necessarilycomes with an associated representation of the double centralizer algebra, compat-ible with the natural homomorphism from A into M(A). This question motivatedthe research leading to [9] as a preparation for the present paper, and, as it turnsout, such results can be obtained. For example, if the algebra A has a bounded leftapproximate identity, and a non-degenerate bounded representation of A is given,then there is an associated bounded representation of the left centralizer algebraMl(A) which is compatible with the natural homomorphism from A into Ml(A),with similar results for right and double centralizer algebras.3 If we want to applythis in our situation, then we need to show that (A oα G)R has a bounded leftapproximate identity. For crossed product C∗-algebras, this is of course automatic,but in the present case it is not. Thus it becomes necessary to establish this inde-pendently, and indeed (Aoα G)R has a bounded approximate left identity if A hasone, with similar right and two-sided results. As an extra complication, since therepresentations of A under consideration are now not necessarily contractive any-more, and the group need not act isometrically, it becomes necessary, with the futureapplications in Section 2.9 in mind, to keep track of the available upper bounds forthe various maps as they are constructed during the process. For this, in turn, oneneeds an explicit upper bound for bounded left and right approximate identities in(A oα G)R. It is for these reasons that Section 2.4 on approximate identities in(Aoα G)R and their bounds, which is superfluous for crossed product C∗-algebras,is a key technical interlude in the present paper.

After that, once we know that (Aoα G)R has a left bounded approximate iden-tity, we can let the left centralizer algebra Ml((A oα G)R) take over the role thatthe double centralizer algebra has for crossed product C∗-algebras. Given a non-degenerate bounded representation T of (Aoα G)R, we can now find a compatible

3Theorem 2.6.1 contains a summary of what is needed in the present paper.


non-degenerate bounded representation of Ml((Aoα G)R), and on composing thiswith existing homomorphisms of the algebra and the group intoMl((AoαG)R), weobtain a pair (π, U) of representations. The continuity and covariance properties areeasily established, as is the non-degeneracy of π, but as compared to the situation forcrossed product C∗-algebras, a complication arises again. Indeed, since in that caseR consists of all non-degenerate involutive covariant representations of (A,G, α) inHilbert spaces, and an involutive T yields and involutive π and unitary U , the pair(π, U) is automatically in R, and is therefore certainly R-continuous. For Banachalgebra dynamical systems this need not be the case, and the norm estimates inour bookkeeping, although useful in Section 2.9, provide no rescue: one needs anindependent proof to show that (π, U) as obtained from T is R-continuous. Oncethis has been done, it is not overly complicated anymore to show that the associatedbounded representation (π o U)R of (A oα G)R (which can then be defined) is Tagain. By keeping track of invariant closed subspaces and bounded intertwining op-erators during the process, and also considering the involutive context at little extracost, the basic correspondence in Theorem 2.8.1 has then finally been established.

With this in place, and also the norm estimates from our bookkeeping available,it is easy so give applications in special situations. This is done in the final section,where we formulate, amongst others, the results for group Banach algebras BX (G)already mentioned above. We then also see that the basic representation theoreticalcorrespondence for “the” C∗-crossed product as commonly associated with a C∗-dynamical system is an instance of a more general correspondence (Theorem 2.9.3),valid for C∗-algebras of crossed product type associated with an involutive (Defi-nition 2.2.10) Banach algebra dynamical systems (A,G, α), provided that, for allε > 0, A has a (1 + ε)-bounded approximate left identity.

This paper is organized as follows.

In Section 2.2 we establish the necessary basic terminology and collect somepreparatory technical results for the sequel. Some of these can perhaps be consideredto be folklore, but we have attempted to make the paper reasonably self-contained,especially since the basics for a general Banach algebra and Banach space situationare akin, but not identical, to those for C∗-algebras and Hilbert spaces, and lesswell-known. At the expense of a little extra verbosity, we have also attempted tobe as precise as possible, throughout the paper, by including the usual conditions,such as (strong) continuity or (in the case of algebras) non-degeneracy of represen-tations, only when they are needed and then always formulating them explicitly,thus eliminating the need to browse back and try to find which (if any) conventionapplies to the result at hand. There are no such conventions in the paper. It wouldhave been convenient to assume from the very start that, e.g., all representationsare (strongly) continuous and (in case of algebras) non-degenerate, but it seemedcounterproductive to do so.

Section 2.3 contains the construction of the crossed product and its basic prop-erties. The ingredients are a given Banach algebra dynamical system (A,G, α) anda uniformly bounded class R of continuous covariant representations thereof.

Section 2.4 contains the existence results and bounds for approximate identities


in (Aoα G)R. As explained above, this is a key issue which need not be addressedin the case of crossed product C∗-algebras.

Section 2.5 is concerned with the easiest part of the representation theory asconsidered in this paper: the passage from R-continuous covariant representationsof (A,G, α) to bounded representations of (AoαG)R. We have included results aboutpreservation of invariant closed subspaces, bounded intertwining operators and non-degeneracy. In this section, two homomorphisms iA and iG of, respectively, A and Ginto End (Cc(G,A)) make their appearance, which will later yield homomorphismsiRA and iRG into the left centralizer algebraMl((AoαG)R), as needed to construct acovariant representation of (A,G, α) from a non-degenerate bounded representationof (AoαG)R. With the involutive case in mind, anti-homomorphism jA and jG intoEnd (Cc(G,A)) are also considered.

Section 2.6 on centralizer algebras starts with a review of part of the results from[9], and then, after establishing a separation property to be used later (Proposi-tion 2.6.2), continues with the study of more or less canonical (anti-)homomorphismsof A and G into the left, right or double centralizer algebra of (A oα G)R. These(anti-)homomorphisms, such as iRA and iRG already alluded to above are based on the(anti-)homomorphisms from Section 2.5.

Section 2.7 contains the most involved part of the representation theory: the pas-sage from non-degenerate bounded representations of (AoαG)R to non-degenerateR-continuous covariant representations of (A,G, α). At this point, if A has abounded left approximate identity, then Sections 2.4 and 2.6 provide the necessaryingredients. If T is a non-degenerate bounded representation of (A oα G)R, thenthere is a compatible non-degenerate bounded representation T ofMl((AoαG)R),and one thus obtains a representation T iRA of A and a representation T iRG of G.The main hurdle, namely to construct any representations of A and G at all fromT , has thus been taken, but still some work needs to be done to take care of theremaining details.

Section 2.8 contains, finally, the bijection between the non-degenerateR-continu-ous covariant representations of (A,G, α) and the non-degenerate bounded repre-sentations of (A oα G)R, valid if A has a bounded left approximate identity andR consists of non-degenerate continuous covariant representations only. Obtainingthis Theorem 2.8.1 is simply a matter of putting the pieces together. Results aboutpreservation of invariant closed subspaces and bounded intertwining operators arealso included, as is a specialization to the involutive case. For convenience, we havealso included in this section some relevant explicit expressions and norm estimatesas they follow from the previous material.

In Section 2.9 the basic correspondence from Theorem 2.8.1 is applied to varioussituations, including the cases of a trivial algebra and of a trivial group. Whereas anapplication of this theorem to the case of a trivial algebra does lead to non-trivialresults about group Banach algebras, as discussed earlier in this Introduction, itdoes not give optimal results for a trivial group. In that case, the machinery of thepresent paper is, in fact, largely superfluous, but for the sake of completeness wehave nevertheless included a brief discussion of that case and a formulation of the(elementary) optimal results.


Reading guide. In the discussion above it may have become evident that, whereasthe construction of a Banach algebra crossed product requires modifications of thecrossed product C∗-algebra construction which are fairly natural and easily imple-mented, establishing the desired correspondence at the level of (covariant) repre-sentations is more involved than for crossed product C∗-algebras. As evidence ofthis may serve the fact that Theorem 2.8.1 can, without too much exaggeration, beregarded as the summary of most material preceding it, including some results from[9]. To facilitate the reader who is mainly interested in this correspondence as such,and in its applications in Section 2.9, we have included (references to) the relevantdefinitions in Sections 2.8 and 2.9. We hope that, with some browsing back, thesetwo sections, together with this Introduction, thus suffice to convey how (AoαG)R

is constructed and what its main properties and special cases are.

Perspectives. According to its preface, [51] can only cover part of what is cur-rently known about crossed products of C∗-algebras in one volume. Although thetheory of crossed products of Banach algebras is, naturally, not nearly as well de-veloped as for C∗-algebras, it is still true that more can be said than we felt couldreasonably be included in one research paper. Therefore, in [22] we will continue thestudy of these algebras. We plan to include (at least) a characterization of (AoαG)R

by a universal property in the spirit of [51, Theorem 2.61], as well as a detailed dis-cussion of L1-algebras. As mentioned above, L1(G) is isometrically isomorphic toa crossed product as constructed in the present paper, and the well-known linkbetween its representation theory and that for G follows from our present results.We will include this, as a special case of similar results for L1(G,A) with twistedconvolution. Also, we will then consider natural variations on the bijection theme:suppose that one has, say, a uniformly bounded class R of pairs (π, U), where π is anon-degenerate continuous anti-representation of A, U is a strongly continuous anti-representation of G, and the pair (π, U) is anti-covariant, is it then possible to findan algebra of crossed product type, the non-degenerate bounded anti-representationsof which correspond bijectively to the R-continuous pairs (π, U) with the propertiesas just mentioned? It is not too difficult to relate these questions to the results inthe present paper, albeit sometimes for a closely related alternative Banach algebradynamical system, and it seems quite natural to consider this matter, since examplesof such R are easy to provide. Once this has been done, we will also be able to inferthe basic relation [24, Proposition 2.1] between L1(G)-bimodules and G-bimodulesfrom the results in the present paper. As mentioned above, we also plan to considerMorita equivalence and imprimitivity theorems, but that may have to wait until an-other time. The same holds for crossed products of Banach algebras in the contextof positive representations on Banach lattices.4

4As a preparation, positivity issues have already been taken into account in [9].

2.2. PRELIMINARIES 27

2.2 Preliminaries

In this section we introduce the basic definitions and notations, and establish somepreliminary results. We start with a few general notions.

If G is a group, then e will be its identity element. If G is a locally compact group,then we fix a left Haar measure µ on G, and denote integration of a function ψ withrespect to this Haar measure by

∫Gψ(s) ds. We let ∆ : G→ (0,∞) denote the mod-

ular function, so for f ∈ Cc(G) and r ∈ G we have ([51, Lemma 1.61, Lemma 1.67])

∆(r)

∫G

f(sr) ds =

∫G

f(s) ds,

∫G

∆(s−1)f(s−1) ds =

∫G

f(s) ds.

If X is a normed space, we denote by B(X) the normed algebra of boundedoperators on X. We let Inv(X) denote the group of invertible elements of B(X). IfA is a normed algebra, we write Aut+(A) for its group of bounded automorphisms.

A neighbourhood of a point in a topological space is a set with that point asinterior point. It is not necessarily open.

Throughout this paper, the scalar field can be either the real or the complexnumbers.

2.2.1 Group representations

Definition 2.2.1. A representation U of a group G on a normed space X is a grouphomomorphism U : G→ Inv(X).

Note that there is no continuity assumption, which is actually quite convenientduring proofs. For typographical reasons, we will write Us rather than U(s), fors ∈ G.

Lemma 2.2.2. Let X be a non-zero Banach space and U be a strongly continuousrepresentation of a topological group G on X. Then for every compact set K ⊂ Gthere exist a constant MK > 0 such that, for all r ∈ K,

1

MK≤ ‖Ur‖ ≤MK .

Proof. For fixed x ∈ X, the map r 7→ Urx is continuous, so the set Urx : r ∈ Kis compact and hence bounded. By the Banach-Steinhaus Theorem, there existsM ′K > 0 such that ‖Ur‖ ≤M ′K for all r ∈ K. Since, for r ∈ K,

1 = ‖idX‖ ≤ ‖Ur−1‖ ‖Ur‖ ≤M ′K−1 ‖Ur‖ ,

MK = max(M ′K ,M′K−1) is as required.

If U is a strongly continuous representation of a topological group G on a Banachspace X, then the natural map from G×X to X is separately continuous. Actually,it is automatically jointly continuous, according to the next result.


Proposition 2.2.3. Let U be a strongly continuous representation of the locallycompact group G on the Banach space X. Then the map (r, x)→ Urx from G×Xto X is continuous.

Proof. We may assume that X is non-zero. Fix (r0, x0) ∈ G × X and let ε > 0.There exists a neighbourhood V of r0 such that, for all r ∈ V , ‖Urx0−Ur0x0‖ < ε/2.We may assume that V is compact, and then Lemma 2.2.2 yields an MV > 0 suchthat ‖Ur‖ ≤ MV for all r ∈ V . Therefore, if r ∈ V and ‖x− x0‖ < ε/(2MV ), wehave

‖Urx− Ur0x0‖ ≤ ‖Urx− Urx0‖+ ‖Urx0 − Ur0x0‖

< MV ·ε

2MV+ε

2

= ε.

Corollary 2.2.5 below, and notably its second statement, will be used repeatedlywhen showing that a representation of a locally compact group is strongly continu-ous. The following lemma is a preparation.

Lemma 2.2.4. Let G and H be two groups with a topology such that right multi-plication is continuous in both groups, or such that left multiplication is continuousin both groups. Let U : G → H be a homomorphism. Then U is continuous if andonly if it is continuous at e.

Proof. Assume that right multiplication is continuous in both groups. Let U be ahomomorphism which is continuous at e and let (ri) ∈ G be a net converging tor ∈ G. Then rir

−1 → e by the continuity of right multiplication by r−1 in G, andso

Uri = Urir−1Ur → Ur,

where the continuity of right multiplication by Ur in H is used in the last step. Thecase of continuous left multiplication is proved similarly, writing Uri = UrUr−1ri .

Corollary 2.2.5. Let G be a group with a topology such that right or left multi-plication is continuous. Let X be a Banach space and suppose U : G → Inv(X)is a representation of G on X. Then U is a strongly continuous representation ifand only U is strongly continuous at e. If U is uniformly bounded on some neigh-bourhood of e, and Y ⊂ X is a dense subset of X, then U is a strongly continuousrepresentation if and only if r 7→ Ury is continuous at e for all y ∈ Y .

Proof. The first part follows from Lemma 2.2.4 and the fact that multiplication inB(X) equipped with the strong operator topology is separately continuous. Thesecond part is an easy consequence of the first.

If X is a Hilbert space, then the ∗-strong operator topology is the topologyon B(X) generated by the seminorms T 7→ ‖Tx‖ + ‖T ∗x‖, with x ∈ X. A net


(Ti) converges ∗-strongly to T if and only if both Ti → T strongly and T ∗i → T ∗

strongly. This topology is stronger than the strong operator topology and weakerthan the norm topology, and multiplication is continuous in this topology on uni-formly bounded subsets of B(X).

Remark 2.2.6. If U is a unitary representation, then the decomposition of r 7→ U∗ras r 7→ r−1 7→ U−1

r = U∗r shows that U is strongly continuous if and only if U is∗-strongly continuous.

2.2.2 Algebra representations

Definition 2.2.7. A representation π of an algebra A on a normed space X is analgebra homomorphism π : A → B(X). The representation π is non-degenerate ifπ(A) ·X := span π(a)x : a ∈ A, x ∈ X is dense in X.

Note that it is not required that π is unital if A has a unit element, nor that πis (norm) bounded if A is a normed algebra.

Remark 2.2.8. If A is a normed algebra with a bounded left approximate identity(ui), and π is a bounded representation of A on the Banach space X, then it is easyto verify that π is non-degenerate if and only if π(ui)→ idX in the strong operatortopology.

The following result, which will be used in the context of covariant representa-tions, follows readily using Remark 2.2.8.

Lemma 2.2.9. Let A be a normed algebra with a bounded approximate left identity,and let π be a bounded representation of A on a Banach space X.

(i) If π is non-degenerate and Z ⊂ X is an invariant subspace, then the restrictedrepresentation of A to Z is non-degenerate.

(ii) There is a largest invariant subspace such that the restricted representation ofA to it is non-degenerate. This subspace is closed. In fact, it is π(A) ·X.

2.2.3 Banach algebra dynamical systems and covariant rep-resentations

We continue by defining the notion of a dynamical system in our setting.

Definition 2.2.10. A normed (resp. Banach) algebra dynamical system is a triple(A,G, α), where A is a normed (resp. Banach) algebra5, G is a locally compactHausdorff group, and α : G → Aut+(A) is a strongly continuous representation ofG on A. The system is called involutive when the scalar field is C, A has a boundedinvolution and αs is involutive for all s ∈ G.

5If A is an algebra, then we do not assume that it is unital, nor that, if it is a unital normedalgebra, the identity element has norm 1.


From Proposition 2.2.3 we see that, for a Banach algebra dynamical system(A,G, α), the canonical map (s, a) → αs(a) is continuous from G × A to A. Thisfact has as important consequence that a number of integrands in the sequel arecontinuous vector valued functions on G, and we mention one of these explicitly forfuture reference.

Lemma 2.2.11. Let (A,G, α) be a Banach algebra dynamical system, let s ∈ G andf ∈ C(G,A). Then the map r 7→ αr

(f(r−1s)

)from G to A is continuous.

Indeed, this map is the composition of the maps r 7→ (r, f(r−1s)) from G toG×A and the canonical map from G×A to A.

Next we define our main objects of interest, the covariant representations.

Definition 2.2.12. Let (A,G, α) be a normed algebra dynamical system, and letX be a normed space. Then a covariant representation of (A,G, α) on X is a pair(π, U), where π is a representation of A on X and U is a representation of G on X,such that for all a ∈ A and s ∈ G,

π(αs(a)) = Usπ(a)U−1s .

The covariant representation (π, U) is called continuous if π is norm bounded andU is strongly continuous, and it is called non-degenerate if π is a non-degeneraterepresentation of A.

If (A,G, α) is a normed algebra dynamical system, then the covariant represen-tation (π, U) of (A,G, α) on X is called involutive if the representation space X is aHilbert space, π is an involutive representation of A and U is a unitary representationof G.

We can use Lemma 2.2.9 to obtain a similar general result for normed dynamicalsystems which, for G = e, specializes to Lemma 2.2.9 again.

Lemma 2.2.13. Let (A,G, α) be a normed algebra dynamical system, where A hasa bounded approximate left identity. Let (π, U) be a covariant representation of(A,G, α) on the Banach space X, and assume that π is bounded.

(i) If (π, U) is non-degenerate and Z is a subspace which is invariant under bothπ(A) and U(G), then the restricted covariant representation of (A,G, α) to Zis non-degenerate.

(ii) There is a largest subspace which is invariant under both π(A) and U(G) suchthat the restricted covariant representation of (A,G, α) to it is non-degenerate.This subspace is closed. In fact, it is π(A) ·X.

Proof. The first part follows directly from the first part of Lemma 2.2.9. As forthe second part, the second part of Lemma 2.2.9 shows that any subspace which isinvariant under both π(A) and U(G) is contained in π(A) ·X. It also yields that πrestricted to this space is a non-degenerate representation of A, hence we need only


show that it is invariant under U(G). As to this, if y = π(a)x, where a ∈ A andx ∈ X, then, for r ∈ G, using the covariance,

Ury = Urπ(a)x = π(αr(a))Urx ∈ π(A) ·X.

By continuity, this implies that π(A) ·X is invariant under Ur, for all r ∈ G.

We conclude this subsection with some terminology about intertwining operators.Let A be an algebra, and let G be a group. Suppose that X and Y are two Banachspaces, and that π : A → B(X) and ρ : A → B(Y ) are two representations of A.Then a bounded operator Φ : X → Y is said to be a bounded intertwining operatorbetween π and ρ if ρ(a) Φ = Φ π(a), for all a ∈ A. A bounded intertwiningoperator between two group representations is defined similarly. If (A,G, α) is anormed dynamical system, and (π, U) and (ρ, V ) are two covariant representationson Banach spaces X and Y , respectively, then a bounded operator Φ : X → Yis called an intertwining operator for these covariant representations, if Φ is anintertwining operator for π and ρ, as well as for U and V .

2.2.4 Cc(G,X)

We will frequently work with functions in Cc(G,X), where X is a Banach space.The next lemma, for the proof of which we refer to [51, Lemma 1.88], shows thatthese functions are uniformly continuous.

Lemma 2.2.14. Let G be a locally compact Hausdorff group and let X be a Banachspace. If f ∈ Cc(G,X) and ε > 0, then there exists a neighbourhood V of e ∈ Gsuch that either one of sr−1 ∈ V or s−1r ∈ V implies

‖f(s)− f(r)‖ < ε.

Remark 2.2.15. In this paper we will sometimes refer to the so-called induc-tive limit topology on Cc(G,X). In these cases, we will be concerned with nets(fi) ∈ Cc(G,X) that converge to f ∈ Cc(G,X), in the sense that (fi) is eventuallysupported in some fixed compact set K ⊂ G, and that (fi) converges uniformly to fon G. As explained in [51, Remark 1.86] and [36, Appendix D.2], such a net is con-vergent in the inductive limit topology, but the converse need not be true. However,it is true that a map from Cc(G,X), supplied with the inductive limit topology, toa locally convex space is continuous precisely when it carries nets which converge inthe above sense to convergent nets. We will use this fact in the sequel.

The algebraic tensor product Cc(G) ⊗ A can be identified with a subspace ofCc(G,A), and the following approximation result will be used on several occasions.We refer to [51, Lemma 1.87] for the proof, from which a part of the formulation inthe version below follows.

Lemma 2.2.16. Let G be a locally compact group, and let X be Banach space. IfX0 is a dense subset of X, then Cc(G) ⊗ X0 is a dense subset of Cc(G,X) in the


inductive limit topology. In fact, it is even true that there exists a sequence (fn) inCc(G) ⊗ X0, with all supports contained in a fixed compact subset of G and whichconverges uniformly to f on G, which implies that fn → f in the inductive limittopology of Cc(G,X).

2.2.5 Vector valued integration

For vector-valued integration in Banach spaces, we base ourselves on an integraldefined by duality. The pertinent definition, as well as the existence, are containedin the next result, for the proof of which we refer to [42, Theorem 3.27] or [51,Lemma 1.91].

Theorem 2.2.17. Let G be a locally compact group, and let X be a Banach spacewith dual space X ′. Then there is a linear map f 7→

∫Gf(s)ds from Cc(G,X) to X

which is characterized by⟨∫G

f(s) ds, x′⟩

=

∫G

〈f(s), x′〉 ds, ∀f ∈ Cc(G,X),∀x′ ∈ X ′. (2.2.1)

Remark 2.2.18. If X and Y are Banach spaces, it follows easily that boundedoperators from X to Y can be pulled through the integral of the above theorem.If X has a bounded involution this can also be pulled through the integral, sincea bounded involution is a bounded conjugate linear map which can be viewed as abounded operator from X to the conjugate Banach space of X.

For F ∈ Cc(G×G,X), it is shown in [51, Proposition 1.102] that, if one integratesout one variable, the resulting function is in Cc(G,X). Applying continuous linearfunctionals, it then follows easily, analogously to the proof of [51, Proposition 1.105],that for such functions F the vector-valued version of Fubini’s theorem is valid.

The integral from Theorem 2.2.17 enables us to integrate compactly supportedstrongly (and ∗-strongly) continuous operator-valued functions (recall that unitaryrepresentations are ∗-strongly continuous by Remark 2.2.6).

Proposition 2.2.19. Let X be a Banach space, let G be a locally compact group,and let ψ : G→ B(X) be compactly supported and strongly continuous. Define∫

G

ψ(s) ds :=

[x 7→

∫G

ψ(s)x ds

], (2.2.2)

where the integral on the right hand side is the integral from Theorem 2.2.17. Then∫Gψ(s) ds ∈ B(X), and ∥∥∥∥∫

G

ψ(s)

∥∥∥∥ ds ≤ ∫G

‖ψ(s)‖ ds. (2.2.3)

If T,R ∈ B(X), then

T

∫G

ψ(s) dsR =

∫G

Tψ(s)Rds. (2.2.4)


Furthermore, if X is a Hilbert space and ψ is ∗-strongly continuous, then(∫G

ψ(s) ds

)∗=

∫G

ψ(s)∗ ds. (2.2.5)

Proof. By applying elements of X and functionals we obtain (2.2.4), while (2.2.3)follows from applying elements of X and taking norms. As for (2.2.5), let x, y ∈ Xbe arbitrary, then⟨

x,

(∫G

ψ(s) ds

)∗y

⟩=

⟨(∫G

ψ(s) ds

)x, y

⟩=

∫G

〈ψ(s)x, y〉 ds

=

∫G

〈x, ψ(s)∗y〉 ds

=

⟨x,

(∫G

ψ(s)∗ ds

)y

⟩,

where the ∗-strong continuity of ψ ensures that the last line is well defined. Sincethis holds for all x, y ∈ X, (2.2.5) follows.

2.2.6 Quotients

The following standard type result, with a routine proof, will be used many timesover, often implicitly. If (D,σ) and (E, τ) are seminormed spaces, a linear mapT : D → E, is said to be bounded if there exists C ≥ 0 such that τ(Tx) ≤ Cσ(x),for all x ∈ D. The seminorm (which is a norm if τ is a norm) of T , is then definedto be the minimal such C.

Lemma 2.2.20. Let (D,σ) and (E, τ) be seminormed spaces, let D/ ker(σ)σ

be

the completion of D/ ker(σ) in the norm induced by σ, and let E/ ker(τ)τ

be definedsimilarly. Suppose that T : D → E is a bounded linear map. Then T [ker(σ)] ⊂ ker(τ)and, with qσ and qτ denoting the canonical maps, there exists a unique boundedoperator T : D/ ker(σ)

σ→ E/ ker(τ)

τ, such that the diagram

D

qσ

T // E

qτ

D/ ker(σ)

σ

T

// E/ ker(τ)τ

(2.2.6)

is commutative. The norm of T then equals the seminorm of T . In particular,if (E, τ) is a Banach space, then T 7→ T is a Banach space isometry between the

bounded operators from D into E and the bounded operators from D/ ker(σ)σ

intoE.


If, in addition, (D,σ) is a seminormed algebra, E is a Banach algebra, and T

is a bounded algebra homomorphism, then ker(σ) is a two-sided ideal, D/ ker(σ)σ

is a Banach algebra, and T is a bounded algebra homomorphism. In particular, ifX is a Banach space and T : D → B(X) is a bounded representation, then T is a

bounded representation of D/ ker(σ)σ

. In this case, T is non-degenerate if and only

if T is non-degenerate, a closed subspace of X is invariant for T if and only if it isinvariant for T , and if Y is a Banach space, Φ ∈ B(X,Y ) and S : D → B(Y ) is a

bounded representation, then Φ intertwines T and S if and only if Φ intertwines Tand S.

Alternatively, if, in addition, D is an algebra with an involution, σ is a C∗-seminorm, E is a Banach algebra with a (possibly unbounded) involution, and T isa bounded involutive algebra homomorphism, then ker(σ) is a self-adjoint two-sided

ideal, D/ ker(σ)σ

is a C∗-algebra, and T is a bounded involutive algebra homomor-phism.

2.3 Crossed product: construction and basic prop-erties

Let (A,G, α) be a Banach algebra dynamical system, and letR be a class of (possiblydegenerate) continuous covariant representations of (A,G, α) on Banach spaces. Wewill construct a Banach algebra, denoted (AoαG)R, which deserves to be called thecrossed product associated with (A,G, α) andR, and establish some basic properties.As will become clear from the discussion below, an additional condition on R isneeded (see Definition 2.3.1), which is automatic, hence “not visible”, in the case ofcrossed product C∗-algebras. In later sections, we will also require the elements ofR to be non-degenerate, but for the moment this is not necessary.

We now start with the construction. Analogously to the C∗-algebra case, thisconstruction is based on the vector space Cc(G,A), as follows.

Let f, g ∈ Cc(G,A). As a consequence of Lemma 2.2.11, the function

(s, r) 7→ f(r)αr(g(r−1s))

is in Cc(G×G,A). Therefore, if s ∈ G, Remark 2.2.18 implies that

[f ∗ g](s) :=

∫G

f(r)αr(g(r−1s)) dr

is a well-defined element of A, and that the thus defined map f ∗ g : G → A,the twisted convolution product of f and g, is in Cc(G,A). The associativity ofthis product on Cc(G,A) is easily shown using Fubini, and thus Cc(G,A) has thestructure of an associative algebra. An easy computation shows that supp (f ∗ g) iscontained in supp (f) · supp (g).

Furthermore, if (A,G, α) is an involutive Banach algebra dynamical system, thenthe formula

f∗(s) := ∆(s−1)αs(f(s−1)∗) (2.3.1)

2.3. CONSTRUCTION AND BASIC PROPERTIES 35

defines an involution on Cc(G,A), so that Cc(G,A) becomes an involutive algebra.The proof of this fact relies on a computation as in [51, page 48], which, as thereader may verify, is valid again because the involution on A is bounded and hencecan be pulled through the integral by Remark 2.2.18.

Our next step is to find an algebra seminorm on Cc(G,A). It is here that asubstantial difference with the construction as in [51] for crossed products associatedwith C∗-algebras occurs, leading to Definition 2.3.1.

To start with, assume that (π, U) is a continuous covariant representation ina Banach space X. For f ∈ Cc(G,A), the function s 7→ π(f(s))Us is stronglycontinuous from G into B(X) by continuity of multiplication in the strong operatortopology on uniformly bounded subsets. Therefore we can define

π o U(f) :=

∫G

π(f(s))Us ds, (2.3.2)

where the integral on the right-hand side is as in (2.2.2). Note that if (π, U) is invo-lutive, U is ∗-strongly continuous by Remark 2.2.6, so s 7→ π(f(s))Us is ∗-stronglycontinuous by continuity of multiplication in the ∗-strong operator topology on uni-formly bounded subsets, hence the involution can be pulled through the integral by(2.2.5). Therefore the computations in the proof of [51, Proposition 2.23] are valid,and they show that πoU , called the integrated form of (π, U), is a representation ofCc(G,A), and that it is involutive if (A,G, α) is involutive and (π, U) is an involutivecontinuous covariant representation.

With the construction as in [51] as a model, the natural way to construct anormed algebra from the associative algebra Cc(G,A), given a collection R of (possi-bly degenerate) continuous covariant representations on Banach spaces of the Banachalgebra dynamical system (A,G, α), is then as follows. For a continuous covariantrepresentation (π, U) and f ∈ Cc(G,A), we define σ(π,U)(f) := ‖π o U(f)‖. Sinceπ o U is a representation of Cc(G,A), the map σ(π,U) : Cc(G,A) → [0,∞) is analgebra seminorm on Cc(G,A). Moreover, if (π, U) is an involutive continuous co-variant representation, then π o U is involutive and hence σ(π,U) is a C∗-seminormon Cc(G,A).

Lemma 2.2.2 shows that U is bounded on compact sets K by constants MK(U),and so we obtain the estimate

σ(π,U)(f) =

∥∥∥∥∫G

π(f(s))Us ds

∥∥∥∥ ≤ ‖π‖Msupp (f)(U) ‖f‖L1(G,A) . (2.3.3)

Now in the case of C∗-algebra dynamical systems, one takes the supremum over allsuch C∗-seminorms corresponding to (non-degenerate) involutive continuous covari-ant representations (π, U), and one defines the corresponding crossed product as thecompletion of Cc(G,A) with respect to this seminorm (which can then be shownto be a norm). This is meaningful: since the constants ‖π‖ and Msupp (f)(U) in(2.3.3) are then always equal to 1, regardless of the choice of (π, U), the supremumis, indeed, pointwise finite. In general situations, when one wants to construct, ina similar way, a crossed product associated with a given class R of continuous co-variant representations, this supremum over the class need no longer be pointwise


finite. Furthermore, even if the supremum is well-defined, this supremum seminormneed not be a norm on Cc(G,A). The solution to these problems is, obviously, toconsider only classes R such that there is a uniform bound for π and Msupp (f)(U)in (2.3.3), as (π, U) ranges over R, and to use a quotient in the construction, as inLemma 2.2.20. This leads to the following definitions.

Definition 2.3.1. Let (A,G, α) be a Banach algebra dynamical system, and supposeR is a class of continuous covariant representations of (A,G, α). Then R is calleduniformly bounded if there exist a constant C ≥ 0 and a function ν : G → [0,∞),which is bounded on compact subsets of G, such that, for all (π, U) in R, ‖π‖ ≤ Cand ‖Ur‖ ≤ ν(r), for all r ∈ G.

If R is a non-empty uniformly bounded class of continuous covariant representa-tions, we let CR = sup(π,U)∈R ‖π‖ be the minimal such C, and we let

νR(r) := sup(π,U)∈R

‖Ur‖ , (2.3.4)

where r ∈ G, be the minimal such ν.6

If (A,G, α) is involutive, then R is said to be involutive if (π, U) is involutive forall (π, U) ∈ R.

Definition 2.3.2. Let (A,G, α) be a Banach algebra dynamical system, and supposeR is a non-empty uniformly bounded class of continuous covariant representations.Then we define the algebra seminorm σR on Cc(G,A) by

σR(f) := sup(π,U)∈R

‖π o U(f)‖ ,

for f ∈ Cc(G,A), and we let the corresponding crossed product (A oα G)R be the

completion of Cc(G,A)/ ker(σR) in the norm ‖ . ‖R induced by σR. Multiplicationin (Aoα G)R will still be denoted by ∗.

Remark 2.3.3.

(i) From now on, all representations are assumed to be on Banach spaces ratherthan on normed spaces, since this is needed when integrating.

(ii) Obviously, σR in Definition 2.3.2 is indeed finite, since, as in (2.3.3),

σR(f) ≤ CR(

supr∈supp (f)

νR(r)

)‖f‖1 <∞,

for f ∈ Cc(G,A).

6Since r 7→ ‖Ur‖ is the supremum of continuous functions, it is a lower semicontinuous functionon G, and hence the same holds for νR.

2.3. CONSTRUCTION AND BASIC PROPERTIES 37

(iii) By construction, (Aoα G)R is a Banach algebra. If (A,G, α) and R are bothinvolutive, then the seminorms σ(π,U), for (π, U) ∈ R, are all C∗-seminorms onCc(G,A), and hence the same holds for their supremum σR. As has alreadybeen observed in Lemma 2.2.20, this implies that (A oα G)R is then a C∗-algebra.

If (A,G, α) is a Banach algebra dynamical system, and R is a non-empty uni-formly bounded class of continuous covariant representations, then the corresponding

quotient map from Lemma 2.2.20 will be denoted by qR, rather than qσR

. Hence

qR : Cc(G,A)→ (Aoα G)R

is the quotient homomorphism. Likewise, if E is Banach space, and the linear mapsT : Cc(G,A)→ E and S : Cc(G,A)→ Cc(G,A) are σR-bounded, then their norms

will be denoted by ‖T‖R and ‖S‖R, and the corresponding bounded operators fromLemma 2.2.20 will be denoted by TR and SR, with norms

∥∥TR∥∥ = ‖T‖R and∥∥SR∥∥ = ‖S‖R. Hence TR : (AoαG)R → E and SR : (AoαG)R → (AoαG)R aredetermined by

TR(qR(f)) = T (f), SR(qR(f)) = qR(S(f)), (2.3.5)

for all f ∈ Cc(G,A).If (π, U) ∈ R is a continuous covariant representation in the Banach space X,

then πoU : Cc(G,A)→ B(X) is certainly σR bounded, with norm at most 1. Hencethere is a corresponding contractive representation (π o U)R : (Aoα G)R → B(X)of the Banach algebra (Aoα G)R in X, determined by

(π o U)R(qR(f)) = π o U(f), (2.3.6)

for all f ∈ Cc(G,A). If (A,G, α) and R are involutive, and X is the Hilbert repre-sentation space for (π, U) ∈ R, then (πoU)R : (AoαG)R → B(X) is an involutiverepresentation of the C∗-algebra (AoαG)R in the Hilbert space X. It is contractiveby construction, although this is of course also automatic.

Suppose that R is a uniformly bounded class of continuous covariant represen-tations. By construction, we have, for f ∈ Cc(G,A),∥∥qR(f)

∥∥R = σR(f) = sup(π,U)∈R

‖π o U(f)‖ = sup(π,U)∈R

∥∥(π o U)R(qR(f))∥∥ .

For later use, we establish that this formula for the norm in (AoαG)R extends fromqR(Cc(G,A)) to the whole crossed product. The separation property that is imme-diate from it, will later find a parallel for the left centralizer algebraMl((AoαG)R)of (AoαG)R in Proposition 2.6.2, under the extra conditions that A has a boundedleft approximate identity and thatR consists of non-degenerate continuous covariantrepresentations only.


Proposition 2.3.4. Let (A,G, α) be a Banach algebra dynamical system and R anon-empty uniformly bounded class of continuous covariant representations. Then,for all c ∈ (Aoα G)R,

‖c‖R = sup(π,U)∈R

∥∥(π o U)R(c)∥∥ .

In particular, the representations (π o U)R, for (π, U) ∈ R, separate the points of(Aoα G)R.

Proof. Let c ∈ (A oα G)R. The contractivity of (π o U)R, for (π, U) ∈ R, yields

sup(π,U)∈R∥∥(π o U)R(c)

∥∥ ≤ ‖c‖R.

As for the other inequality, let ε > 0. Choose an f ∈ Cc(G,A) such that∥∥c− qR(f)∥∥R < ε/3, and pick (π, U) ∈ R such that ‖π o U(f)‖ >

∥∥qR(f)∥∥R− ε/3.

Then ∥∥(π o U)R(c)∥∥ ≥ ∥∥(π o U)R(qR(f))

∥∥− ∥∥(π o U)R(c− qR(f))∥∥

≥ ‖π o U(f)‖ −∥∥c− qR(f)

∥∥R≥∥∥qR(f)

∥∥R − ε

3− ε

3

≥ ‖c‖R − ε

3− 2ε

3.

Therefore, sup(π,U)∈R∥∥(π o U)R(c)

∥∥ > ‖c‖R − ε for all ε > 0, as desired.

We will now proceed to show that qR(Cc(G)⊗A) is dense in (Aoα G)R, whichwill obviously be convenient in later proofs. We start with a lemma which is of someinterest in itself.

Lemma 2.3.5. Let (A,G, α) be a Banach algebra dynamical system.

(i) If R is a non-empty uniformly bounded class of continuous covariant repre-sentations, then qR : Cc(G,A) → (A oα G)R is continuous in the inductivelimit topology of Cc(G,A). That is, if (fi) ∈ Cc(G,A) is a net, eventuallysupported in a compact set and converging uniformly to f ∈ Cc(G,A) on G,then σR(fi − f)→ 0.

(ii) If (π, U) is a continuous covariant representation, then π o U is continuousin the inductive limit topology. That is, if (fi) ∈ Cc(G,A) is a net, eventuallysupported in a compact set and converging uniformly to f ∈ Cc(G,A) on G,then ‖π o U(fi)− π o U(f)‖ → 0.

Proof. (i) It suffices to prove the case when f = 0. Let K be a compact set andi0 an index such that fi is supported in K for all i ≥ i0. Let M denote an

2.4. APPROXIMATE IDENTITIES 39

upper bound for νR on K. Then, for (π, U) ∈ R and i ≥ i0,

‖π o U(fi)‖ =

∥∥∥∥∫G

π(fi(s))Us ds

∥∥∥∥≤∫K

‖π‖ ‖fi(s)‖M ds

≤ CRMµ(K) ‖fi‖∞ .

It follows that, for i ≥ i0,

σR(fi) = sup(π,U)∈R

‖π o U(fi)‖ ≤ CRMµ(K) ‖fi‖∞ .

Hence σR(fi)→ 0.

(ii) This follows from (i) by taking R = (π, U), since then σR(f) = ‖π o U(f)‖.

Corollary 2.3.6. Let (A,G, α) be a Banach algebra dynamical system and R anon-empty uniformly bounded class of continuous covariant representations. ThenqR(Cc(G)⊗A) is dense in (Aoα G)R.

Proof. By [51, Lemma 1.87], Cc(G) ⊗ A is dense in Cc(G,A) in the inductivelimit topology. The above lemma therefore implies that qR(Cc(G) ⊗ A) is densein qR(Cc(G,A)). Since the latter is dense in (Aoα G)R by construction, the resultfollows.

2.4 Approximate identities

In this section we are concerned with the existence of a bounded approximate leftidentity in (A oα G)R. This is a key issue in the formalism, as the existence of abounded left approximate identity will allow us to apply Theorem 2.6.1 later on topass from (non-degenerate) representations of (A oα G)R to representations of theleft centralizer algebra of (Aoα G)R, from which, in turn, we will be able to obtaina continuous covariant representation of (A,G, α).

If (A oα G)R happens to be a C∗-algebra, as is, e.g., the case in [51], then theexistence of a bounded two-sided approximate identity is of course automatic, butfor the general case some effort is needed to show that the existence of a boundedleft approximate identity in A implies the similar property for (A oα G)R. Forthe present paper, we need only a bounded left approximate identity, but we alsoconsider an approximate right identity for completeness and future use.

We need two preparatory results. The first one, Lemma 2.4.1, is only relevantfor the case of a bounded approximate right identity.

Lemma 2.4.1. Let (A,G, α) be a Banach algebra dynamical system and suppose Ahas a bounded approximate right identity (ui). Fix an element a ∈ A and a compact


set K ⊂ G. Then for all ε > 0 we can find an index i0 such that, for all i ≥ i0 ands ∈ K,

‖aαs(ui)− a‖ < ε.

Proof. Let M ≥ 1 be an upper bound for (ui). By Lemma 2.2.2 we can choose anupper bound MK > 0 for α on K.

By the continuity of s 7→ s−1 7→ αs−1(a) = α−1s (a), for every s ∈ K there exists

a neighbourhood Ws such that, for all r ∈Ws,

‖α−1r (a)− α−1

s (a)‖ < ε

3MKM.

Choose a finite subcover Ws1 , . . . ,Wsn of K. Then there exists an index i0 such thati ≥ i0 implies

‖α−1sk

(a)ui − α−1sk

(a)‖ < ε

3MK,

for every 1 ≤ k ≤ n. For s ∈ K, choose k such that s ∈Wsk . Then, for all i ≥ i0,

‖aαs(ui)− a‖ =∥∥αs(α−1

s (a)ui − α−1s (a))

∥∥≤MK

(‖α−1

s (a)ui − α−1sk

(a)ui‖+ ‖α−1sk

(a)ui − α−1sk

(a)‖+‖α−1

sk(a)− α−1

s (a)‖)

< MK

(ε

3MKM·M +

ε

3MK+

ε

3MKM

)≤ ε.

Actually, the existence of a bounded approximate left (resp. right) identity in(AoαG)R is inferred from the existence of an suitable approximate left (resp. right)identity in Cc(G,A) in the inductive limit topology. This is the subject of thefollowing theorem.

Theorem 2.4.2. Let (A,G, α) be a Banach algebra dynamical system and let Z bea neighbourhood basis of e of which all elements are contained in a fixed compact set.For each V ∈ Z, take a positive zV ∈ Cc(G) with support contained in V and integralequal to one. Suppose (ui) is a bounded approximate left (resp. right) identity of A.Then the net

(f(V,i)

), where

f(V,i) := zV ⊗ ui,

directed by (V, i) ≤ (W, j) if and only if W ⊂ V and i ≤ j, is an approximate left(resp. right) identity of Cc(G,A) in the inductive limit topology. In fact, for allf ∈ Cc(G,A) the net

(f(V,i) ∗ f

)(resp.

(f ∗ f(V,i)

)) is supported in a fixed compact

set and converges uniformly to f on G.

Proof. Let K be a compact set containing all V in Z, and assume that (ui) isbounded by M > 0.

Since the f(V,i) are all supported in K, all f(V,i) ∗f (resp. f ∗f(V,i)) are supportedin K supp (f) (resp. supp (f)K) for each f ∈ Cc(G)⊗A.


For the approximating property we start with elements of Cc(G) ⊗ A. For thisit is sufficient to consider elementary tensors, so fix 0 6= f = z ⊗ a with z ∈ Cc(G)and a ∈ A.

First we consider the approximate left identity. Suppose that ε > 0 is given. LetMK > 0 be an upper bound for α on K. Then, for s ∈ G,

∥∥[f(V,i) ∗ f ](s)− f(s)∥∥ =

∥∥∥∥∫G

f(V,i)(r)αr(f(r−1s)) dr − f(s)

∥∥∥∥=

∥∥∥∥∫G

zV (r)z(r−1s)uiαr(a) dr − z(s)a∥∥∥∥

=

∥∥∥∥∫G

zV (r)z(r−1s)uiαr(a)− zV (r)z(s)a dr

∥∥∥∥≤∫G

zV (r)∥∥z(r−1s)uiαr(a)− z(s)a

∥∥ dr≤∫

supp (zV )

zV (r)∥∥z(r−1s)uiαr(a)− z(s)uiαr(a)

∥∥ dr+

∫supp (zV )

zV (r) ‖z(s)uiαr(a)− z(s)uia‖ dr

+

∫supp (zV )

zV (r) ‖z(s)uia− z(s)a‖ dr

≤MMK ‖a‖∫

supp (zV )

zV (r)|z(r−1s)− z(s)| dr

+ ‖z‖∞M∫

supp (zV )

zV (r) ‖αr(a)− a‖ dr

+ ‖z‖∞∫

supp (zV )

zV (r) ‖uia− a‖ dr.

By the uniform continuity of z, there exists a neighbourhood U1 of e such that|z(r−1s) − z(s)| < ε/(3MMK ‖a‖), for all r ∈ U1 and s ∈ G. Hence, for all s ∈ G,the first term is less than ε/3 as soon as V ⊂ U1. By the strong continuity of α,there exists a neighbourhood U2 of e such that ‖αr(a)− a‖ < ε/(3M‖z‖∞), for allr ∈ U2. Hence, for all s ∈ G, the second term is less than ε/3 as soon as V ⊂ U2.There exists an index i0 such that the third term is less than ε/3 for all i ≥ i0 andall V ∈ Z. Choose V0 ∈ Z such that V0 ⊂ U1 ∩ U2. Then, if (V, i) ≥ (V0, i0), wehave

∥∥f(V,i) ∗ f(s)− f(s)∥∥ < ε for all s ∈ G as required.

The approximate right identity is somewhat more involved. Suppose that ε > 0is given. Let K1 be a compact set containing all V in Z, as well as the supports ofall f ∗ f(V,i) and f . Let MK1K−1 > 0 be an upper bound for α on K1K

−1. Since


f(s) ∗ f(V,i)(s)− f(s) = 0 for s /∈ K1, we assume that s ∈ K1, and then∥∥[f ∗ f(V,i)](s)− f(s)∥∥

=

∥∥∥∥∫G

f(r)αr(f(V,i)(r−1s)) dr − f(s)

∥∥∥∥=

∥∥∥∥∫G

z(r)azV (r−1s)αr(ui) dr − z(s)a∥∥∥∥

=

∥∥∥∥∫G

z(sr)zV (r−1)aαsr(ui) dr − z(s)a∥∥∥∥

=

∥∥∥∥∫G

∆(r−1)z(sr−1)zV (r)aαsr−1(ui)− zV (r)z(s)a dr

∥∥∥∥≤∫G

zV (r)∥∥∆(r−1)z(sr−1)aαsr−1(ui)− z(s)a

∥∥ dr≤∫

supp (zV )

zV (r)∥∥∆(r−1)z(sr−1)aαsr−1(ui)− z(sr−1)aαsr−1(ui)

∥∥ dr+

∫supp (zV )

zV (r)∥∥z(sr−1)aαsr−1(ui)− z(s)aαsr−1(ui)

∥∥ dr+

∫supp (zV )

zV (r) ‖z(s)aαsr−1(ui)− z(s)a‖ dr

≤ ‖z‖∞ ‖a‖MK1K−1M

∫supp (zV )

zV (r)|∆(r−1)− 1| dr

+ ‖a‖MK1K−1M

∫supp (zV )

zV (r)|z(sr−1)− z(s)| dr

+ ‖z‖∞∫

supp (zV )

zV (r) ‖aαsr−1(ui)− a‖ dr.

By the continuity of ∆ there exists a neighbourhood U1 of e such that

|∆(r−1)− 1| < ε/(3‖z‖∞ ‖a‖MK1K−1M),

for all r ∈ U1. Hence, for all s ∈ K1, the first term is less than ε/3 as soon asV ⊂ U1. By the uniform continuity of z, there exists neighbourhood U2 of e suchthat |z(sr−1) − z(s)| < ε/(3 ‖a‖MKK−1

1M), for all r ∈ U2 and s ∈ G. Hence, for

all s ∈ K1, the second term is less than ε/3 as soon as V ⊂ U2. An application ofLemma 2.4.1 to the compact set K1K

−1 shows that there exists an index i0 suchthat ‖aαsr−1(ui)− a‖ < ε/(3‖z‖∞), for all i ≥ i0, s ∈ K1 and r ∈ K. Hence, forall s ∈ K1, the third term is less than ε/3 if i ≥ i0. Choose V0 ∈ Z such thatV0 ⊂ U1 ∩ U2. Then, if (V, i) ≥ (V0, i0), we have

∥∥[f ∗ f(V,i)](s)− f(s)∥∥ < ε for all

s ∈ K1, and hence for all s ∈ G, as required.

We now pass from Cc(G) ⊗ A to Cc(G,A), using that Cc(G) ⊗ A is uniformlydense in Cc(G,A) (a rather weak consequence of [51, Lemma 1.87]).


We start with the left approximate identity. Let f ∈ Cc(G,A) and ε > 0 begiven. For arbitrary g ∈ Cc(G,A) and s ∈ G we have∥∥[f(V,i) ∗ f ](s)− f(s)

∥∥≤∥∥[f(V,i) ∗ (f − g)](s)

∥∥+∥∥[f(V,i) ∗ g](s)− g(s)

∥∥+ ‖g(s)− f(s)‖

=

∥∥∥∥∫G

zV (r)uiαr((f − g)(r−1s)) dr

∥∥∥∥+∥∥[f(V,i) ∗ g](s)− g(s)

∥∥+ ‖g(s)− f(s)‖

≤MMK‖f − g‖∞∫

supp (zV )

zV (r) dr +∥∥[f(V,i) ∗ g](s)− g(s)

∥∥+ ‖g − f‖∞

≤ (MMK + 1)‖f − g‖∞ +∥∥f(V,i) ∗ g − g

∥∥∞ .

The cited density yields g ∈ Cc(G) ⊗ A such that the first term is less than ε/2.By the first part of the proof, there exists an index (V0, i0) such that the secondterm is less than ε/2 for all (V, i) ≥ (V0, i0). Therefore ‖f(V,i) ∗ f − f‖∞ < ε for all(V, i) ≥ (V0, i0).

As for the approximate right identity, let f ∈ Cc(G,A) and ε > 0 be given. Asabove, we let K1 be a compact set containing all V in Z, as well as the supportsof all f ∗ f(V,i) and f , and choose an upper bound MK1K−1 > 0 for α on K1K

−1.Let NK−1 be an upper bound for ∆ on K−1. Then, for arbitrary g ∈ Cc(G,A) ands ∈ K1, we have∥∥f ∗ f(V,i)(s)− f(s)

∥∥≤∥∥[(f − g) ∗ f(V,i)](s)

∥∥+∥∥[g ∗ f(V,i)](s)− g(s)

∥∥+ ‖g(s)− f(s)‖

=

∥∥∥∥∫G

(f − g)(r)zV (r−1s)αr(ui) dr

∥∥∥∥+∥∥[g ∗ f(V,i)](s)− g(s)

∥∥+ ‖g(s)− f(s)‖

=

∥∥∥∥∥∫

supp (zV )

∆(r−1)(f − g)(sr−1)zV (r)αsr−1(ui) dr

∥∥∥∥∥+∥∥[g ∗ f(V,i)](s)− g(s)

∥∥+ ‖g(s)− f(s)‖

≤ NK−1‖f − g‖∞MK1K−1M

∫supp (zV )

zV (r) dr

+∥∥[g ∗ f(V,i)](s)− g(s)

∥∥+ ‖g − f‖∞≤ (NK−1MK1K−1M + 1)‖f − g‖∞ + ‖g ∗ f(V,i) − g‖∞.

As above, there exists an index (V0, i0) such that, for all (V, i) ≥ (V0, i0),

‖f ∗ f(V,i)(s)− f(s)‖ < ε

for all s ∈ K1. Since this is trivially true for s /∈ K1, we are done.

After these preparations we can now establish that (A oα G)R has a boundedapproximate left identity if A has one. We keep track of the constants rather pre-cisely, since the upper bound for the norms of a bounded approximate left identity


of (A oα G)R will enter the picture naturally when considering the relation be-tween (non-degenerate) continuous covariant representations of (A,G, α) and (non-degenerate) bounded representations of (A oα G)R later on, cf. Remark 2.8.4 andSection 2.9. Therefore, before we prove the result on the approximate identities, letus introduce the relevant constant.

Definition 2.4.3. Let (A,G, α) be a Banach algebra dynamical system and R anon-empty uniformly bounded class of continuous covariant representations. LetνR : G → [0,∞) be defined as in (2.3.4). Let Z be a neighbourhood basis of e ofwhich all elements are contained in a fixed compact set, and define

NR = infV ∈Z

supr∈V

νR(r) <∞. (2.4.1)

Note that, since all V in Z are contained in a fixed compact set, and νR isbounded on compacta, NR is indeed finite. Furthermore, this definition of NR

does not depend on the choice of Z. To see this, let Z1 and Z2 be two neigh-bourhood bases as in the theorem. For every V1 ∈ Z1, there exists V2 ∈ Z2

such that V2 ⊂ V1, and then supr∈V2νR(r) ≤ supr∈V1

νR(r). This implies thatinfV ∈Z2 supr∈V ν

R(r) ≤ infV ∈Z1 supr∈V νR(r), and the independence of the choice

obviously follows.The constant NR can be viewed as lim supr→e ν

R(r).

Theorem 2.4.4. Let (A,G, α) be a Banach algebra dynamical system, where A hasan M -bounded approximate left (resp. right) identity (ui), and let R be a non-emptyuniformly bounded class of continuous covariant representations. Let ε > 0, andchoose a neighbourhood V0 of e with compact closure such that

NR ≤ supr∈V0

νR(r) ≤ NR + ε.

Let Z be a neighbourhood basis of e of which all elements are contained in V0. Foreach V ∈ Z, let zV ∈ Cc(G) be a positive function with support contained in V andintegral equal to one. Define f(V,i) := zV ⊗ ui, for each V ∈ Z and each index i.

Then the associated net(qR(f(V,i))

)as above is a CRM(NR + ε)-bounded ap-

proximate left (resp. right) identity of (Aoα G)R.If V0 satisfies NR = supr∈V0

νR(r), then(qR(f(V,i))

)is CRMNR-bounded.

Proof. We will prove the left version, the right version is similar.If V ∈ Z and i is an index then we find that, for (π, U) ∈ R,

∥∥π o U(f(V,i))∥∥ =

∥∥∥∥∫V0

zV (s)π(ui)Us ds

∥∥∥∥ (2.4.2)

≤ ‖π‖ ‖ui‖ supr∈V‖Ur‖

∫V

zV (s) ds

≤ CRM supr∈V0

νR(r).

2.5. REPRESENTATIONS: FROM (A,G, α) TO (Aoα G)R 45

Hence σR(qR(f(V,i))) = σR(f(V,i)) ≤ CRM supr∈V0νR(r) ≤ CRM(NR + ε), as

desired. To show that(qR(f(V,i))

)is actually an approximate left identity, we start

by noting that, according to Theorem 2.4.2, f(V,i) ∗ f → f in the inductive limittopology on Cc(G,A), for all f ∈ Cc(G,A). Therefore, Lemma 2.3.5 implies thatqR(f(V,i)) ∗ qR(f)→ qR(f), for all f ∈ Cc(G,A). Since we have already established

that(qR(f(V,i))

)is uniformly bounded in (A oα G)R, an easy 3ε-argument shows

that the net is indeed a left approximate identity of (Aoα G)R.As for the second part, if V0 and Z are as indicated and V ⊂ V0 is in Z, then a

computation as in (2.4.2) yields∥∥π o U(f(V,i))∥∥ ≤ CRM sup

r∈V0

νR(r) ≤ CRM(NR + ε),

hence σR(qR(f(V,i))) ≤ CRM(NR + ε). This computation with ε = 0 shows thefinal remark of the theorem.

For convenience we introduce the following notation.

Definition 2.4.5. Let A be a normed algebra with a bounded approximate left(resp. right) identity. ThenMAl (resp.MAr ) denotes the infimum of the upper boundsof all approximate left (resp. right) identities. If (A,G, α) is a Banach algebra dy-namical system, with A having a bounded left (resp. right) approximate identity, andR is a non-empty uniformly bounded class of continuous covariant representations,

then we will write MRl (resp. MRr ), rather than M(AoαG)R

l

(resp. M

(AoαG)R

r

).

Corollary 2.4.6. Let (A,G, α) be a Banach algebra dynamical system, where A hasa bounded approximate left (resp. right) identity (ui), and let R be a non-emptyuniformly bounded class of continuous covariant representations. Then (A oα G)R

has a bounded approximate left (resp. right) identity, and

MRl ≤ CRMAl N

R,

MRr ≤ CRMAr N

R.

Proof. We prove the left version, the right version being similar. If ε > 0, then Ahas an MA

l + ε approximate left identity, so by the above theorem (AoαG)R has aCR(MA

l +ε)(NR+ε)-bounded approximate left identity, and the result follows.

2.5 Representations: from (A,G, α) to (Aoα G)R

Our principal interest lies in the relation between a non-empty uniformly boundedclass R of continuous covariant representations of a Banach algebra dynamical sys-tem (A,G, α), and the bounded representations of the associated crossed product(Aoα G)R. In this section, we study the easiest part of this relation, which is con-cerned with passing from suitable continuous covariant representations of (A,G, α) to


σR-bounded representations of Cc(G,A), and subsequently to bounded representa-tions of (AoαG)R. The other way round, i.e., passing from bounded representationsof (AoαG)R to continuous covariant representations of (A,G, α), is more involved,and will be taken up in Section 2.7, after the preparatory Section 2.6. At that point,non-degeneracy of representations will become essential, but for the present sectionthis is not necessary yet.

Above, we wrote “suitable” representations, because there are more continuouscovariant representations yielding bounded representations of the crossed product,than just those used to construct that crossed product (which yield contractive ones).The relevant terminology is introduced in the following definition.

Definition 2.5.1. Let (A,G, α) be a Banach algebra dynamical system, and let Rbe a non-empty uniformly bounded class of continuous covariant representations.A covariant representation (π, U) of (A,G, α) in a Banach space X is called R-continuous, if it is continuous, and the homomorphism

π o U : Cc(G,A)→ B(X)

is σR-bounded.

Remark 2.5.2. It is clear that a continuous covariant representation (π, U) of(A,G, α) is R-continuous if and only if πoU is continuous as an operator from thespace Cc(G,A), equipped with the topology induced by the seminorm σR, to B(X),equipped with the norm topology.

By Lemma 2.2.20, an R-continuous covariant representation yields a boundedrepresentation of the Banach algebra (AoαG)R, determined by (2.3.5), which gives(2.3.6) again:

(π o U)R(qR(f)) = π o U(f), (2.5.1)

for f ∈ Cc(G,A). Then ‖π o U‖ = ‖π o U‖R. If, in addition, (A,G, α), R and(π, U) are involutive, then (πoU)R is an involutive representation of the C∗-algebra(AoαG)R. Of course, returning to the not necessarily involutive case, the continuouscovariant representations in R are certainly R-continuous, and the correspondingrepresentations of (Aoα G)R are contractive.

Later, in Proposition 2.7.1, we will be able to show that, if R is a non-emptyuniformly bounded class of continuous covariant representations and if A has abounded left approximate identity, the assignment (π, U) 7→ (π o U)R is injectiveon the non-degenerate R-continuous covariant representations. For the momentwe are interested in the preservation of non-degeneracy, the set of closed invariantsubspaces and the Banach space of intertwining operators under this map. As a firststep, we consider these issues for the assignment (π, U) 7→ πoU , for a still arbitrarycontinuous covariant representation (π, U). In order to do this, we introduce fourmaps which will be very useful later on as well. Two of these (jA and jG below) willonly be needed in the involutive case.


Proposition 2.5.3. Let (A,G, α) be a Banach algebra dynamical system. Definemaps iA, jA : A→ End (Cc(G,A)) and iG, jG : G→ End (Cc(G,A)) by

[iA(a)f ](s) := af(s), [jA(a)f ](s) := f(s)αs(a), (2.5.2)

[iG(r)f ](s) := αr(f(r−1s)), [jG(r)f ](s) := ∆(r−1)f(sr−1).

for a ∈ A, r ∈ G and f ∈ Cc(G,A). Then iA and iG and homomorphisms and jAand jG are anti-homomorphisms. If (π, U) is a continuous covariant representationof (A,G, α), then, for a ∈ A, r ∈ G, and f ∈ Cc(G,A),

π o U(iA(a)f) = π(a) π o U(f), π o U(jA(a)f) = π o U(f) π(a), (2.5.3)

π o U(iG(r)f) = Ur π o U(f), π o U(jG(r)f) = π o U(f) Ur.

It is actually true that iA and iG map into the left centralizers of Cc(G,A) andthat jA and jG map into the right centralizers of Cc(G,A). The former will beshown during the proof of Proposition 2.6.4 and the proof of the latter is similar, cf.Proposition 2.6.5.

Proof. It is easy to check that the maps are (anti-)homomorphisms. Let a ∈ A andf ∈ Cc(G,A) and let (π, U) be a continuous covariant representation, then using thecovariance,

π o U(jA(a)f) =

∫G

π[(jA(a)f)(s)]Us ds

=

∫G

π(f(s))π(αs(a))Us ds

=

∫G

π(f(s))Usπ(a) ds

= π o U(f) π(a),

and for r ∈ G we obtain

π o U(iG(r)f) =

∫G

π[(iG(r)f)(s)]Us ds

=

∫G

π[αr(f(r−1s))]Us ds

=

∫G

π[αr(f(s))]UrUs ds

=

∫G

Urπ(f(s))Us ds

= Ur π o U(f).

The other computations are similar and will be omitted.

Before we continue we need a preparatory result, in which the version for jA willnot be applied immediately, but will be useful later on.


Lemma 2.5.4. Let (A,G, α) be a Banach algebra dynamical system with a boundedapproximate left (resp. right) identity (ui), and let f ∈ Cc(G,A)). Then iA(ui)f(resp. jA(ui)f) converges to f in the inductive limit topology of Cc(G,A).

Proof. Starting with the left version, we note that Lemma 2.2.16 implies easily thatit is sufficient to prove the statement for elementary tensors. So let f = z ⊗ a bean elementary tensor in Cc(G) ⊗ A. Then [iA(ui)f ](s) = z(s)uia, which convergesuniformly to z(s)a = f(s) on G.

As to the right version, again Lemma 2.2.16, when combined with the observationthat the operators αs ∈ B(A) are uniformly bounded as s ranges over a compact sub-set of G, implies that it is sufficient to prove the statement for f = z⊗a ∈ Cc(G)⊗A.Let ε > 0. Then, for s ∈ G,

‖[jA(ui)f ](s)− f(s)‖ = ‖z(s)aαs(ui)− z(s)a‖ ≤ |z(s)| ‖aαs(ui)− a‖ .

For s /∈ supp (z), the right hand side is zero, for all i. Lemma 2.4.1 shows that thereexists an index i0 such that, for all i ≥ i0, the right hand side is less than ε, for alls ∈ supp (z), and i ≥ i0. Hence ‖jA(ui)f − f‖∞ → 0. Therefore, jA(ui)f → f inthe inductive limit topology of Cc(G,A).

Proposition 2.5.5. Let (A,G, α) be a Banach algebra dynamical system and let(π, U) be a continuous covariant representation of (A,G, α) on the Banach space X.

(i) If (π, U) is non-degenerate, then π o U is a non-degenerate representation ofCc(G,A). If A has a bounded approximate left identity, the converse also holds.

(ii) If Y is a closed subspace of X which is invariant for both π and U , then Y isinvariant for π o U .

(iii) If Y is a Banach space, (ρ, V ) a continuous covariant representation on Y andΦ : X → Y a bounded intertwining operator between (π, U) and (ρ, V ), thenΦ is a bounded intertwining operator between π o U and ρ o V . If (π, U) isnon-degenerate, the converse also holds.

(iv) If (A,G, α) and (π, U) are involutive, then so is π o U .

Proof. (i) Suppose 0 6= x ∈ X is of the form x = π(a)y, and let ε > 0. Bythe strong continuity of U there exists a neighbourhood V of e such thats ∈ V implies that ‖Usy − y‖ < ε/ ‖π(a)‖. Let z ∈ Cc(G) be nonnegativewith compact support contained in V and with integral equal to 1. Define


f := z ⊗ a ∈ Cc(G,A), then

‖π o U(f)y − x‖ =

∥∥∥∥∫G

π(f(s))Usy ds−∫G

z(s)x ds

∥∥∥∥≤∫G

z(s) ‖π(a)Usy − π(a)y‖ ds

≤∫G

z(s) ‖π(a)‖ ‖Usy − y‖ ds

≤∫

supp (z)

z(s) ‖π(a)‖ ε

‖π(a)‖ds = ε.

This implies that π o U(Cc(G,A)) ·X ⊃ π(A) ·X. Therefore, if π is non-degenerate, then so is π o U .

For the converse, let (ui) be a bounded approximate left identity of A. ByRemark 2.2.8 we have to show that π(ui)x → x, for all x ∈ X. By theboundedness of π and (ui) and an easy 3ε-argument, it is sufficient to showthis for x in a dense subset of X. For this we choose π o U(Cc(G,A)) · X,which is dense in X by assumption. Let f ∈ Cc(G,A) and y ∈ X, then using(2.5.3),

π(ui)π o U(f)y = π o U(iA(ui)f)y → π o U(f)y

by Lemma 2.5.4 and the continuity of π o U in the inductive limit topology(Lemma 2.3.5). Hence π(ui)x→ x for all x ∈ πoU(Cc(G,A)) ·X by linearity.

(ii) If Y is a closed subspace invariant for both π and U , then it is immediatefrom the properties of our vector-valued integral that Y is also invariant underπ o U(Cc(G,A)).

(iii) Let Φ : X → Y be a bounded intertwining operator between (π, U) and (ρ, V ).Then for x ∈ X and f ∈ Cc(G,A) we have

Φ π o U(f) = Φ ∫G

π(f(s))Us ds

=

∫G

Φπ(f(s))Us ds

=

∫G

ρ(f(s)) ΦUs ds

=

∫G

ρ(f(s))Vs Φ ds

=

∫G

ρ(f(s))Vs ds Φ

= ρo V (f) Φ.

Conversely, suppose that Φ : X → Y is a bounded intertwining operator forU o π and V o ρ and that (π, U) is non-degenerate. For elements of X of the


form π o U(f)x, where x ∈ X and f ∈ Cc(G,A), we obtain for r ∈ G, using(2.5.3),

[Φ Ur](π o U(f)x) = [Φ Ur (π o U)(f)]x

= [Φ (π o U)(iG(r)f)]x

= [(ρo V )(iG(r)f) Φ]x

= [Vr (ρo V )(f) Φ]x

= [Vr Φ (π o U)(f)]x

= [Vr Φ](π o U(f)x).

By (i), π o U is non-degenerate, and so Φ Ur and Vr Φ agree on a densesubset of X and hence are equal. Similarly we obtain that Φ π(a) = ρ(a) Φfor all a ∈ A.

(iv) This has been shown in Section 2.3, following (2.3.2).

Together with Lemma 2.2.20 the above proposition immediately leads to mostthe following.

Theorem 2.5.6. Let (A,G, α) be a Banach algebra dynamical system, and let Rbe a non-empty uniformly bounded class of continuous covariant representations.Consider the assignment (π, U) → (π o U)R from the R-continuous covariant rep-resentations of (A,G, α) to the bounded representations of (Aoα G)R.

(i) If (π, U) is non-degenerate, then (π o U)R is a non-degenerate representationof (Aoα G)R. If A has a bounded approximate left identity, the converse alsoholds.

(ii) If Y is a closed subspace of X which is invariant for both π and U , then Yis invariant for (π o U)R. If (π, U) is non-degenerate and A has a boundedapproximate left identity, the converse also holds.

(iii) If Y is a Banach space, (ρ, V ) a continuous covariant representation on Y andΦ : X → Y a bounded intertwining operator between (π, U) and (ρ, V ), then Φis a bounded intertwining operator between (π o U)R and (ρo V )R. If (π, U)is non-degenerate, the converse also holds.

(iv) If (A,G, α), R, and (π, U) are involutive, then so is (π o U)R.

Furthermore, for a general Banach dynamical system, ‖π o U‖R =∥∥(π o U)R

∥∥,for each R-continuous covariant representation (π, U) of (A,G, α).

Proof. All that has to be shown is that if Y is invariant for (πoU)R, (π, U) is non-degenerate and A has a bounded approximate left identity, then Y is invariant for(π, U). Under these assumptions, the first part of the theorem shows that (πoU)R

is non-degenerate. By Theorem 2.4.4, (A oα G)R has a bounded approximate left

2.6. CENTRALIZER ALGEBRAS 51

identity, so by Lemma 2.2.9, (π o U)R|Y is non-degenerate. Using the density ofqR(Cc(G,A)) in (Aoα G)R, we obtain that

π o U(Cc(G,A)) · Y = (π o U)R(qR(Cc(G,A))

)· Y

is dense in Y . Now for a ∈ A, r ∈ G, y ∈ Y and f ∈ Cc(G,A) by (2.5.3),

π(a) π o U(f)y = π o U(iA(a)f)y ∈ YUr π o U(f)y = π o U(iG(r)f)y ∈ Y,

so π(a) and Ur map a dense subset of Y into Y , which proves the claim.

2.6 Centralizer algebras

The passage from non-degenerate bounded representations of (A oα G)R to con-tinuous covariant representations of (A,G, α) in Section 2.7 will be obtained usingthe left centralizer algebra of (A oα G)R. This will be done in Proposition 2.7.1below, and it consists of two steps. The idea is to first construct, by general means,a bounded representation of the left centralizer algebra of (Aoα G)R from a givennon-degenerate bounded representation of (AoαG)R, and next to compose this newrepresentation with covariant homomorphisms (to be constructed below) of A andG into this left centralizer algebra, thus obtaining (at least algebraically) a covariantrepresentations of the group and the algebra.

In the present section, which is a preparation for the next, we start by recallingthe basic general theorem which underlies the first step in the above procedure.This will make it obvious why it is so important that (A oα G)R has a boundedleft approximate identity if A has one, something which is—as observed before–automatic in the C∗-case, but not in the general setting. Next we construct thehomomorphisms needed for the second step. We also include some results for thedouble centralizer algebra of (AoαG)R; these will be needed for the involutive caseonly.

Commencing with representations of a general normed algebra and its centralizeralgebras, we let A be a normed algebra: the results below will be applied with theBanach algebra A = (Aoα G)R. We let Ml(A) ⊂ B(A) denotes the unital normedalgebra of left centralizers of A, i.e., the algebra of bounded operators L : A → Acommuting with all right multiplications, or equivalently, satisfying L(a)b = L(ab)for all a, b ∈ A. Every a ∈ A determines a left centralizer by left multiplication,and we let λ : A → Ml(A) denotes the corresponding homomorphism. Likewise,the algebraMr(A) ⊂ B(A) denotes the unital normed algebra of right centralizers,i.e., the algebra of operators R : A → A commuting with all left multiplications,or equivalently, satisfying R(ab) = aR(b) for all a, b ∈ A, and ρ : A → Mr(A)denotes the canonical anti-homomorphism. The unital normed algebra of doublecentralizers of A is denoted by M(A) and consists of pairs (L,R), where L is a leftcentralizer and R is a right centralizer, such that aL(b) = R(a)b for all a, b ∈ A.Multiplication in M(A) is defined by (L1, R1)(L2, R2) = (L1L2, R2R1) and the


norm by ‖(L,R)‖M(A) = max(‖L‖ , ‖R‖). We let φl : M(A) → Ml(A) denote

the contractive unital homomorphism (L,R) 7→ L, and δ : A → M(A) denote thehomomorphism a 7→ (λ(a), ρ(a)).

If L is invertible inMl(A) and R is invertible inMr(A), then (L,R) is invertiblein M(A) with inverse (L,R)−1 = (L−1, R−1). If A has a bounded involution, thenfor L ∈ Ml(A) the map L∗ : A → A defined by L∗(a) := (L(a∗))∗ is a rightcentralizer, and for R ∈ Mr(A) the map R∗ defined by R∗(a) := (R(a∗))∗ is a leftcentralizer. Furthermore (L∗)∗ = L and (R∗)∗ = R. As a consequence, the map(L,R) 7→ (R∗, L∗) is a bounded involution on M(A).

Obviously, if A is a Banach algebra, then so are Ml(A), Mr(A), and M(A).In the following theorem we collect a few results from [9, Remark 2.2, Theo-

rem 4.1 and Theorem 4.5]. The constant MAl in it is was defined in Definition 2.4.5as the infimum of the upper bounds of all approximate left identities. The theoremimplies, in particular, that, given a non-degenerate bounded Banach space represen-tation of a normed algebra with a bounded approximate left identity, there existsa unique representation (which is then automatically bounded and non-degenerate)of its left centralizer algebra which is compatible with the canonical homomorphismλ : A → Ml(A). This is a crucial step in our approach, and it should be thoughtof as the analogue of extending a representation of a C∗-algebra to its multiplieralgebra.

Theorem 2.6.1. Let A be a normed algebra with a bounded approximate left iden-tity, and let X be a Banach space.

If T : A → B(X) is a non-degenerate bounded representation, then there exists aunique homomorphism T :Ml(A)→ B(X) such that the diagram

A T //

λ $$IIIIIIIIII

δ

B(X)

M(A)φl

//Ml(A)

T

OO(2.6.1)

is commutative. All maps in the diagram are bounded homomorphisms, and T isunital. One has

∥∥T∥∥ ≤ MAl ‖T‖, which implies∥∥T φl∥∥ ≤ MAl ‖T‖. In particular,

T and T φl are non-degenerate bounded representations of Ml(A) and M(A) onX.

The image T (A) is a left ideal in T (Ml(A)). In fact, if L ∈Ml(A) and a ∈ A,then

T (L) T (a) = T (L(a)). (2.6.2)

If (ui) is any bounded approximate left identity of A and if L ∈Ml(A), then forx ∈ X we have

T (L)x = limiT (L(ui))x. (2.6.3)

In particular, the set of closed invariant subspaces of T coincides with the set ofclosed invariant subspaces of T , and if S : A → B(X) is another non-degenerate


bounded representation, then the set of bounded intertwining operators of T and Scoincides with the set of bounded intertwining operators of T and S.

If in addition A has a bounded involution, X is a Hilbert space and T is involutive,then T φl is involutive.

If, returning to our original context, (A,G, α) is a Banach algebra dynamicalsystem, and R is a non-empty uniformly bounded class of continuous covariant rep-resentations, then each (π, U) in R, being obviously R-continuous, yields a bounded(even contractive) representation (π o U)R of (A oα G)R, and if (π, U) ∈ R isnon-degenerate, then (π o U)R is non-degenerate as well, by Theorem 2.5.6. If,in addition, A has a bounded approximate left identity, then (A oα G)R has abounded approximate left identity by Corollary 2.4.6, hence Theorem 2.6.1 pro-vides a bounded representation (π o U)R ofMl((AoαG)R). These representations

(π o U)R, for (π, U) ∈ R, are used in the following result, which is a parallel of theseparation property in Proposition 2.3.4.

Proposition 2.6.2. Let (A,G, α) be a Banach algebra dynamical system, whereA has a bounded approximate left identity, and let R be a non-empty uniformlybounded class of non-degenerate continuous covariant representations. Then thenon-degenerate bounded representations (π o U)R ofMl((AoαG)R), for (π, U) ∈ R,separate the points of Ml((Aoα G)R).

Proof. Let L ∈ Ml((A oα G)R) be such that (π o U)R(L) = 0, for all (π, U) ∈ R.Then, for arbitrary c ∈ (AoαG)R, the combination of Proposition 2.3.4 and (2.6.2)shows that

‖L(c)‖R = sup(π,U)∈R

∥∥(π o U)R(L(c))∥∥

= sup(π,U)∈R

∥∥∥(π o U)R(L) (π o U)R(c)∥∥∥

= 0.

Hence L = 0.

We continue our preparation for the representation theory in the next sectionby investigating a particular continuous covariant representation of (A,G, α) in(A oα G)R, needed for the second step in the procedure outlined in the beginningof this section. An important feature, in view of Theorem 2.6.1, of this particularcontinuous covariant representations is that the corresponding images of A and Gare contained in the left centralizer algebra Ml((Aoα G)R) of (Aoα G)R, so thatit can be composed with representations of Ml((Aoα G)R) resulting from the firststep. We will now proceed to construct this continuous covariant representations,which is done using the actions of A and G on Cc(G,A), as defined in (2.5.2).

Lemma 2.6.3. Let (A,G, α) be a Banach algebra dynamical system, and let R bea non-empty uniformly bounded class of continuous covariant representations. Let


a ∈ A and r ∈ G. Then the maps

iA(a), iG(r) : (Cc(G,A), σR)→ (Cc(G,A), σR)

are bounded. In fact,

‖iA(a)‖R ≤ sup(π,U)∈R

‖π(a)‖ ≤ CR ‖a‖ ,

and‖iG(r)‖R ≤ νR(r).

Proof. Let a ∈ A. Then, for f ∈ Cc(G,A) and (π, U) ∈ Rr, using (2.5.3) in the firststep, we find that

‖π o U(iA(a)f)‖ = ‖π(a) U o π(f)‖

≤

(sup

(π,U)∈R‖π(a)‖

)(sup

(π,U)∈R‖π o U(f)‖

)

=

(sup

(π,U)∈R‖π(a)‖

)σR(f).

Taking the supremum over (π, U) ∈ R implies the statement concerning iA(a). Thestatement concerning iG(r) follows similarly.

As a consequence of the above proposition and Lemma 2.2.20, the operatorsiA(a) and iG(r) yield bounded operators from (A oα G)R to itself with the samenorm. For typographical reasons, we will denote these elements of B((AoαG)R) byiRA (a) and iRG(r) rather than iA(a)R and iG(r)R. Hence, if a ∈ A and r ∈ G, theniRA (a), iRG(r) ∈ B((Aoα G)R) are determined by

iRA (a)(qR(f)) = qR(iA(a)f), iRG(r)(qR(f)) = qR(iG(r)f) (2.6.4)

for all f ∈ Cc(G,A).In Proposition 2.5.3 we have noted that the maps iA : A → End (Cc(G,A))

and iG : G → End (Cc(G,A)) are homomorphisms. As a consequence of (2.6.4)and the density of qR(Cc(G,A)) in (Aoα G)R, the same is then true for the mapsiRA : A → B((A oα G)R) and iRG : G → B((A oα G)R). Hence we have a pair ofrepresentations (iRA , i

RG) on (Aoα G)R.

Proposition 2.6.4. Let (A,G, α) be a Banach algebra dynamical system, and letR be a non-empty uniformly bounded class of continuous covariant representations.Then (iRA , i

RG), as defined by (2.6.4), is a continuous covariant representation of

(A,G, α) in (A oα G)R. The images iRA (A) and iRG(G) are contained in the leftcentralizer algebra Ml((Aoα G)R) of (Aoα G)R, so we have

iRA : A→Ml((Aoα G)R) ⊂ B((Aoα G)R),

iRG : G→Ml((Aoα G)R) ⊂ B((Aoα G)R).


For the operator norm in B((Aoα G)R) the estimates∥∥iRA (a)∥∥ ≤ sup

(π,U)∈R‖π(a)‖ ≤ CR ‖a‖ ,

where a ∈ A, and ∥∥iRG(r)∥∥ ≤ νR(r),

where r ∈ G, hold.If, in addition, A has a bounded approximate left identity, then (iRA , i

RG) is non-

degenerate.

Although it does not follow from the estimates for the operator norm in Propo-sition 2.6.4, if A has a bounded approximate left identity and all elements of Rare non-degenerate, then it is actually true that (iRA , i

RG) is R-continuous, see The-

orem 2.7.2. Proving this will require some extra effort, and we will only be ableto do so once more information has been obtained about the relation between R-continuous covariant representations of (A,G, α) and bounded representations of(Aoα G)R.

Proof. We start by proving the covariance of (iRA , iRG). For this it is sufficient to show

that the pair (iA, iG) is covariant, i.e., that [iG(r)iA(a)iG(r)−1f ](s) = [iA(αr(a))f ](s)for all f ∈ Cc(G,A), r, s ∈ G, and a ∈ A. Indeed,

[iG(r)iA(a)iG(r)−1f ](s) = αr[(iA(a)iG(r)−1f)(r−1s)]

= αr[aiG(r)−1f(r−1s)]

= αr[aαr−1(f(rr−1s))]

= αr(a)(f(s))

= [iA(αr(a))f ](s).

We continue by showing that the bounded operators iRA (a) and iRG on (AoαG)R

are left centralizers of the Banach algebra (AoαG)R. To see this, let a ∈ A. Then,for f, g ∈ Cc(G,A) and s ∈ G,

[iA(a)(f ∗ g)](s) = a

∫G

f(r)αr(g(r−1s)) dr

=

∫G

af(r)αr(g(r−1s)) dr

= [(iA(a)f) ∗ g](s).

So iA(a) commutes with right multiplication in Cc(G,A). Hence

iRA (a)(qR(f) ∗ qR(g)) = iRA (a)(qR(f ∗ g)) = qR(iA(a)(f ∗ g))

= qR([iA(a)f ] ∗ g) = qR(iA(a)f) ∗ qR(g)

= [iA(a)RqR(f)] ∗ qR(g),


for f, g ∈ Cc(G,A). From the density of qR(Cc(G,A) in (AoαG)R and the bound-edness of iRA (a) it then follows that iRA (a) is a left centralizer of (A oα G)R. As tothe other case, let r ∈ G. Then, for f, g ∈ Cc(G,A) and s ∈ G,

[iG(r)(f ∗ g)](s) = αr([f ∗ g](r−1s))

= αr

(∫G

f(t)αt(g(t−1r−1s)) dt

)=

∫G

αr(f(t))αrt(g((rt)−1s)) dt

=

∫G

αr(f(r−1t))αt(g(t−1s)) dt

= [(iG(r)f) ∗ g](s).

So iG(r) commutes with right multiplication in Cc(G,A). As for iA(a), it followsthat iRG(r) is a left centralizer of (Aoα G)R.

Next, we will show that iRG is strongly continuous. In view of the boundednessof iRG on compact neighbourhoods of e, Corollary 2.2.5 implies that we only haveto show strong continuity of iRG in e on a dense subset of (A oα G)R. By Corol-lary 2.3.6, qR(Cc(G) ⊗ A) is dense in (A oα G)R, and so it is sufficient to showthat σR(iG(ri)f − f) → 0 for all f ∈ Cc(G) ⊗ A, whenever ri → e in G. By lin-earity it is sufficient to consider only elements of the form z ⊗ a with z ∈ Cc(G)and a ∈ A. Therefore, fix z ⊗ a and let ri → e. We may assume that the ri areall contained in a fixed compact set. It is the obvious that the net (iG(ri)(z ⊗ a))is likewise supported in a fixed compact set, so by Lemma 2.3.5 it suffices to showthat [iG(ri)(z ⊗ a)](s)− z(s)a→ 0, uniformly in s. Since

‖[iG(ri)(z ⊗ a)](s)− z(s)a‖ =∥∥z(r−1

i s)αri(a)− z(s)a∥∥

≤∥∥z(r−1

i s)αri(a)− z(r−1i s)a

∥∥+∥∥z(r−1

i s)a− z(s)a∥∥

≤ ‖z‖∞ ‖αri(a)− a‖+∥∥z(r−1

i s)− z(s)∥∥ ‖a‖ ,

this uniform convergence follows from the strong continuity of α and the uniform con-tinuity of z. Together with the discussion preceding the theorem, this concludes theproof that (iRA , i

RG) is a continuous covariant representation of (A,G, α) on (AoαG)R.

If, in addition, A has a bounded approximate left identity (ui), then, for eachf ∈ Cc(G)⊗A, Lemma 2.5.4 shows that iA(ui)f → f in the inductive limit topology.As a consequence, iA(A) · Cc(G) ⊗ A is dense in Cc(G) ⊗ A in the inductive limittopology. By Lemma 2.3.5, iRA (A) · qR(Cc(G)⊗A) = qR(iA(A) ·Cc(G)⊗A) is densein qR(Cc(G)⊗ A). Since the latter is dense in (Aoα G)R by Corollary 2.3.6, iRA isthus seen to be non-degenerate.

The above Proposition 2.6.4 is sufficient for the sequel in the case of generalBanach algebra dynamical systems. In the involutive case, the left centralizer algebraalone is no longer sufficient, because of the lack of an involutive structure. In thatcase, we will use the double centralizer algebra, and in order to establish the result


for the double centralizer algebra that will eventually be used, we first need thefollowing right-sided version of part of the above theorem.

Proposition 2.6.5. Let (A,G, α) be a Banach algebra dynamical system and letR be a non-empty uniformly bounded class of continuous covariant representations.For a ∈ A and r ∈ G, let jA(a) and jG(r) be as in (2.6.4). Then the maps

jA(a), jG(r) : (Cc(G,A), σR)→ (Cc(G,A), σR)

are bounded. Denote the corresponding bounded operators on (A oα G)R by jRA (a)and jRG (r), determined by jRA (a)(qR(f)) = qR(jA(a)(f)), for all f ∈ Cc(G,A), andby jRG (r)(qR(f)) = qR(jG(r)f), for all f ∈ Cc(G,A).

Then jRA : A→ B((AoαG)R) is a bounded anti-representation of A in (AoαG)R,and jRG : G → B((A oα G)R) is a strongly continuous anti-representation of G in(Aoα G)R. The pair (jRA , j

RG ) is anti-covariant in the sense that, for all a ∈ A and

all r ∈ G,jRA (αr(a)) = jRG (r)−1jRA (a)jRG (r).

The images jRA (A) and jRG (G) are contained inMr((AoαG)R), the right centralizeralgebra of (Aoα G)R, so we have

jRA : A→Mr((Aoα G)R) ⊂ B((Aoα G)R),

jRG : G→Mr((Aoα G)R) ⊂ B((Aoα G)R).

For the operator norm in B((Aoα G)R) the estimates∥∥jRA (a)∥∥ ≤ sup

(π,U)∈R‖π(a)‖ ≤ CR ‖a‖ ,

where a ∈ A, and ∥∥jRG (r)∥∥ ≤ νR(r),

where r ∈ G, hold.If, in addition, A has a bounded approximate right identity, then jRA is non-

degenerate.

Proof. The proof is similar to the proof of the corresponding statements in Propo-sition 2.6.4 and the details are therefore omitted.

Proposition 2.6.6. Let (A,G, α) be a Banach algebra dynamical system and let Rbe a non-empty uniformly bounded class of continuous covariant representations ofcontinuous covariant representations. For a ∈ A and r ∈ G, let iRA (a) and iRG(r) beas in Proposition 2.6.4, and let jRA (a) and jRG (r) be as in Proposition 2.6.5. Then((iRA (a), jRA (a)) and (iRG(r), jRG (r)) are both double centralizers of (AoαG)R, and wehave ∥∥(iRA (a), jRA (a))

∥∥ ≤ sup(π,U)∈R

‖π(a)‖ ≤ CR ‖a‖ ,

and ∥∥(iRG(r), jRG (r))∥∥ ≤ νR(r).


Furthermore, the maps a 7→ (iRA (a), jRA (a)) and r 7→ (iRG(r), jRG (r)) are homomor-phisms of A into M((A oα G)R) and of G into M((A oα G)R), respectively, andthe pair ((iRA , j

RA ), (iRG , j

RG )) is covariant in the sense that

(iRA (αr(a)), jRA (αr(a))) = (iRG(r), jRG (r)) · (iRA (a), jRA (a)) · (iRG(r), jRG (r))−1,

for all a ∈ A and all r ∈ G.Moreover, if (A,G, α) and R are involutive, then

(iRA , jRA ) : A→M((Aoα G)R)

is an involutive homomorphism, and (iRG(r), jRG (r))∗ = (iRG(r−1), jRG (r−1)), for allr ∈ G.

Proof. Let a ∈ A and suppose f, g ∈ Cc(G,A). Then the computation, for s ∈ G,

f ∗ (iA(a)g)(s) =

∫G

f(r)αr(iA(a)g(r−1s)) dr

=

∫G

f(r)αr(a)αr(g(r−1s)) dr

= (jA(a)f) ∗ g(s)

shows that (iA(a), jA(a)) is a double centralizer of Cc(G,A). By continuity anddensity, the same holds for (iRA (a), jRA (a)) and (Aoα G)R. Similarly, if r, s ∈ G andf, g ∈ Cc(G,A), then

f ∗ (iG(r)g)(s) =

∫G

f(t)αt((iG(r)g)(t−1s)) dt

=

∫G

f(t)αt(αr(g(r−1t−1s))) dt

=

∫G

f(t)αtr(g((tr)−1s)) dt

=

∫G

∆(r−1)f(tr−1)αt(g(t−1s)) dt

=

∫G

(jG(r)f)(t)αt(g(t−1s)) dt

= (jG(r)f) ∗ g(s)

implies that (iRG(r), jRG (r)) is a double centralizer of (Aoα G)R.The fact that the maps are homomorphisms and the covariance property fol-

low directly from the corresponding statements in Proposition 2.6.4 and Proposi-tion 2.6.5, and the definition of the inverse and the multiplication in the doublecentralizer algebra.

As to the final statement, suppose that (A,G, α) is involutive, and thatR consistsof involutive representations. To show that the homomorphism (iA, jA) from A into

2.7. REPRESENTATIONS: FROM (Aoα G)R TO (A,G, α) 59

M((AoαG)R) is involutive, we have to show that (iRA (a), jRA (a))∗ = (iRA (a∗), jRA (a∗))for all a ∈ A, i.e., that (jRA (a)∗, iRA (a)∗) = (iRA (a∗), jRA (a∗)). Recalling the definitions(2.3.1) and (2.6.4), we find, for f ∈ Cc(G,A) and s ∈ G, that

[jA(a)∗f ] (s) = [jA(a)f∗]∗

(s)

= ∆(s−1)αs

[(jA(a)f∗)(s−1)

∗]= ∆(s−1)αs

[f∗(s−1)αs−1(a)

∗]= ∆(s−1)αs

[∆(s)αs−1(f(s)∗)αs−1(a)∗

]= a∗f(s)

= [iA(a∗)f ](s).

and

[iA(a)∗f ] (s) = [iA(a)f∗]∗

(s)

= ∆(s−1)αs

[(iA(a)f∗)(s−1)

∗]= ∆(s−1)αs

[af∗(s−1)

∗]= ∆(s−1)αs

[a∆(s)αs−1(f(s)∗)∗

]= f(s)αs(a

∗)

= [jA(a∗)f ] (s),

By continuity and density, this implies that (jRA (a)∗, iRA (a)∗) = (iRA (a∗), jRA (a∗)), asdesired.

A similar unwinding of the definitions establishes, by continuity and density, thatiRG(r)∗ = jRG (r−1), for all r ∈ G. Taking adjoints, this implies jRG (r)∗ = iRG(r−1),hence (iRG(r), jRG (r))∗ = (jRG (r)∗, (iRG(r)∗) = (iRG(r−1), jRG (r−1)), for all r ∈ G.

2.7 Representations: from (Aoα G)R to (A,G, α)

As already indicated in the previous section, Theorem 2.6.1 and Proposition 2.6.4provide a means to generate a covariant representation of (A,G, α) from a non-degenerate bounded representation of (A oα G)R, as follows. If A has a boundedapproximate left identity, then the same holds for (A oα G)R, by Corollary 2.4.6,and hence any non-degenerate bounded representation T of (A oα G)R yields abounded representation T of Ml((Aoα G)R), by Theorem 2.6.1. Since, by Propo-sition 2.6.4, the images iRA (A) and iRG(G) are contained in Ml((A oα G)R), thepair of maps (T iRA , T iRG) is meaningfully defined and will then be a covariantrepresentation of (A,G, α), since the covariance requirement is automatically satis-fied as a consequence of the covariance property of (iRA , i

RG), the latter being part

of Proposition 2.6.4. Some natural questions that arise are, e.g., whether this co-variant representation is continuous, and, if so, whether it is R-continuous. We will


now investigate these and related matters, and incorporate some of the results fromSection 2.5 (the passage in the other direction, from R-continuous covariant rep-resentations of (A,G, α) to bounded representations of (A oα G)R) in the process.After that, the proofs of our main results in Section 2.8 will be a mere formality.

Recall from Definition 2.4.5 and Corollary 2.4.6 that MRl denotes the infimumof the upper bounds of the approximate left identities of (Aoα G)R, with estimateMRl ≤ CRMA

l NR, where MA

l denotes the infimum of the upper bounds of theapproximate left identities of A.

Proposition 2.7.1. Let (A,G, α) be a Banach algebra dynamical system, whereA has a bounded approximate left identity, and let R be a non-empty uniformlybounded class of continuous covariant representations. Let (iRA , i

RG) be the continuous

covariant representation of (A,G, α) on (Aoα G)R, as in Proposition 2.6.4.Suppose that T is a non-degenerate bounded representation of (AoαG)R in a Ba-

nach space X, and let T be the associated bounded representation ofMl((AoαG)R)in X, as in Theorem 2.6.1. Then the pair (T iRA , T iRG) is a non-degenerate con-tinuous covariant representation of (A,G, α) in X. For the operator norm on thebounded operators on the representation space the estimates∥∥(T iRA) (a)

∥∥ ≤MRl ‖T‖ sup(π,U)∈R

‖π(a)‖ ≤MRl ‖T‖CR ‖a‖ ,

where a ∈ A, and ∥∥(T iRG) (r)∥∥ ≤MRl ‖T‖ νR(r),

where r ∈ G, hold.If a closed subspace of X is invariant for T , it is invariant for T iRA and

T iRG , and if Y is a Banach space, S : (A oα G)R → B(Y ) a representationand Φ ∈ B(X,Y ) intertwines T and S, then Φ intertwines (T iRA , T iRG) and(S iRA , S iRG).

If, in addition, (A,G, α), R, and T are involutive, then (T iRA , T iRG) is invo-lutive.

Moreover, if, in the not necessarily involutive case, (π, U) is an non-degenerateR-continuous covariant representation of (A,G, α), with corresponding non-degene-rate bounded representation (π o U)R of (Aoα G)R, then(

(π o U)R iRA , (π o U)R iRG)

= (π, U). (2.7.1)

Although it does not follow from the estimates for the operator norm in thetheorem, if all elements of R are non-degenerate, then it is (in analogy with thecontinuous covariant representation (iRA , i

RG) of (A,G, α) in (A oα G)R), actually

true that (T iRA , T iRG) is R-continuous, see Theorem 2.7.3.Note that the final statement of the theorem implies the injectivity of the as-

signment (π, U) → (π o U)R on the non-degenerate R-continuous covariant repre-sentations if A has a bounded approximate left identity, as was already announcedfollowing Definition 2.5.1.


Proof. Let T be a non-degenerate bounded representation of (A oα G)R in theBanach space X. As already remarked preceding the theorem, the definitions

π := T iRA and U := T iRG

are meaningful and provide a covariant representation (π, U) of (A,G, α). We showthat it has the properties as claimed, and start with the bounds for ‖π‖ and ‖Ur‖,for r ∈ G. Let ε > 0, then (A oα G)R has an (MRl + ε)-bounded approximateleft identity. Since Theorem 2.6.1 and Proposition 2.6.4 provide a bound for

∥∥T∥∥,∥∥iRA (a)∥∥, and

∥∥iRG(r)∥∥, we have, for a ∈ A,

‖π(a)‖ ≤∥∥T∥∥∥∥iRA (a)

∥∥≤ (MRl + ε) ‖T‖ sup

(ρ,V )∈R‖ρ(a)‖

≤ (MRl + ε) ‖T‖CR ‖a‖

and, for r ∈ G,

‖Ur‖ ≤∥∥T∥∥∥∥iRG(r)

∥∥ ≤ (MRl + ε) ‖T‖ νR(r).

Since ε > 0 was arbitrary, this establishes the estimates in the theorem.

We have to prove that π is non-degenerate and that U is strongly continuous.Starting with π, by Remark 2.2.8 it has to be shown that π(ui)x→ x for all x ∈ X,where (ui) is a bounded approximate left identity of A. By the boundedness of π,which we already established, and the boundedness of (ui), it is sufficient to establishthis for x in a dense subset of X. Now since T is non-degenerate and qR(Cc(G)⊗A)is dense in (Aoα G)R by Corollary 2.3.6, T (qR(Cc(G)⊗ A)) ·X is dense in X. Solet x ∈ X and f ∈ Cc(G)⊗A, then by (2.6.2) in Theorem 2.6.1,

π(ui)T (qR(f))x = T (iRA (ui))T (qR(f))x

= T [iRA (ui)(qR(f))]x

= T [qR(iA(ui)f)]y

→ T (qR(f))x,

where the last step is by Lemma 2.5.4, Lemma 2.3.5 and the boundedness of T .

Now we turn to the strong continuity of U , Since we have already establishedthat ‖Ur‖ ≤ νR(r), for r ∈ G, and νR is bounded on compact sets, Corollary 2.2.5shows that it is sufficient to show strong continuity of U in e when acting on adense subset of X. For this set we choose T ((Aoα G)R) ·X, which is dense by thenon-degeneracy of T , and then by linearity it is sufficient to show strong continuityin e when acting on elements of the form T (c)y, where c ∈ (Aoα G)R, and y ∈ X.So let x = T (c)y ∈ X, and let ri → e. Then by (2.6.2) in Theorem 2.6.1 we find

Urix = T (iRG(ri))T (c)y = T (iRG(ri)(c))y.


By Proposition 2.6.4, iRG is strongly continuous. Hence, by the continuity of T ,

Urix = T (iRG(ri)(c))y → T (c)y = x,

as required.Suppose Y is a closed invariant subspace ofX for T . By Theorem 2.6.1 T (L)y ∈ Y

for all y ∈ Y and L ∈Ml((AoαG)R). Applying this with L = iA(a) and L = iG(r),for a ∈ A and r ∈ G, shows that Y is invariant for T iA and T iG.

If Y is a Banach space, S a non-degenerate bounded representation of (AoαG)R

in Y and Φ a bounded intertwining operator for T and S, then it follows fromTheorem 2.6.1 that Φ T (L) = S(L) Φ for all L ∈ Ml((A oα G)R). Againapplying this with L = iA(a) and L = iG(r), for a ∈ A and r ∈ G, shows thatΦ [T iA](a) = [S iA](a) Φ and Φ [T iG](r) = [S iG](r) Φ.

Considering the statement on involutions, suppose that, in addition, (A,G, α)and R are both involutive. Let T be an involutive representation of (AoαG)R. ByProposition 2.6.6 the homomorphism (iRA , j

RA ) : A → M((A oα G)R) is involutive

and Theorem 2.6.1 shows that T φl is involutive. Combining these, we obtain that

π = T iRA = T [φl (iRA , j

RA )]

=[T φl

] (iRA , j

RA )

is an involutive representation of A. Finally, if r ∈ G, then using the involutiveproperty of T φl again, as well as Proposition 2.6.6, we see that

U∗r =[T (iRG(r))

]∗=[(T φl

) ((iRG(r), jRG (r))

)]∗=(T φl

) [(iRG(r), jRG (r))∗

]=(T φl

) [(jRG (r)∗, iRG(r)∗)

]=(T φl

)[(iRG(r−1), jRG (r−1))]

= T (iRG(r−1)) = Ur−1 = U−1r .

Hence U is a unitary representation of G, and this completes the proof that the pair(T iRA , T iRG) is involutive.

To conclude with, we consider the final statement on the recovery of an R-continuous covariant representation (π, U) from (π o U)R. Starting with π, leta ∈ A. Then the compatibility equation (2.6.2) in Theorem 2.6.1, when appliedwith L replaced with iRA (a) and a replaced with qR(f), for f ∈ Cc(G,A), yields

(π o U)R(iRA (a)) (π o U)R(qR(f)) = (π o U)R(iRA (a)qR(f))

= π o U(iA(a)f). (2.7.2)

Take an element x ∈ X of the form x = π o U(f)y, with f ∈ Cc(G,A). Using(2.5.3), we find that

π(a)x = π(a) π o U(f)y = π o U(iA(a)f)y.


Combining this with both sides of (2.7.2) acting on y, we see that, for such x,

(π o U)R(iRA (a))x = π(a)x. (2.7.3)

By Proposition 2.5.5 the linear span of elements of the form π o U(f)y, withf ∈ Cc(G,A) and y ∈ X, is dense in X, and therefore (2.7.3) implies that π(a)

and (π o U)R(iRA (a)) are equal.

The proof that (π o U)R(iG(r)) = Ur, for r ∈ G, is similar.

The reconstruction formula (2.7.1) will enable us to complete our results on thecontinuous covariant representation (iRA , i

RG) from Proposition 2.6.4, under the extra

conditions that A has a bounded approximate left identity and that all elements ofR are non-degenerate.

Theorem 2.7.2. Let (A,G, α) be a Banach algebra dynamical system, where A has abounded approximate left identity, and let R be a non-empty uniformly bounded classof non-degenerate continuous covariant representations. Then the non-degeneratecontinuous covariant representation (iRA , i

RG) of (A,G, α) on (AoαG)R from Propo-

sition 2.6.4 is R-continuous, and the associated non-degenerate bounded representa-tion (iRA o iRG)R of (AoαG)R on itself coincides with the left regular representation,and is therefore contractive.

Proof. We start by showing that (iRA , iRG) is R-continuous. If f ∈ Cc(G,A), then

iA(f(s))iG(s) is a left centralizer for all s ∈ G, and hence commutes with all rightmultiplications. By (2.2.4) these right multiplications can be pulled through theintegral, therefore iRA o iRG(f) =

∫GiA(f(s))iG(s) ds commutes with all right multi-

plications as well, hence it is a left centralizer.Let λ denote the left regular representation of (AoαG)R. Then using (2.7.1) in

the fourth step, we find that, for all (π, U) ∈ R,

(π o U)R(iRA o iRG(f)

)= (π o U)R

(∫G

iRA (f(s))iRG(s) ds

)=

∫G

(π o U)R(iRA (f(s))iRG(s)) ds

=

∫G

(π o U)R(iRA (f(s))) · (π o U)R(iRG(s)) ds

=

∫G

π(f(s))Us ds

= π o U(f)

= (π o U)R(qR(f))

= (π o U)R(λ(qR(f))),

where diagram (2.6.1) was used in the final step. By Proposition 2.6.2 the represen-

tations (π o U)R separate the points, and it follows that

iRA o iRG(f) = λ(qR(f)).


Consequently,∥∥iRA o iRG(f)

∥∥ ≤ ‖λ‖ ∥∥qR(f)∥∥R = ‖λ‖σR(f). We conclude that

iRA o iRG is R-continuous.Next we consider the statement that (iRAoiRG)R, the representation of (AoαG)R

on itself, which we now know to be defined as a consequence of the first part of theproof, is the left regular representation. Let f ∈ Cc(G,A). Then, for all (π, U) ∈ R,the above computation shows that

(π o U)R((iRA o iRG)R(qR(f))

)= (π o U)R

(iRA o iRG(f)

)= (π o U)R(λ(qR(f))),

so again by the point-separating property of the representations (π o U)R it followsthat (iRA o iRG)R(qR(f)) = λ(qR(f)). By continuity and density, the statementfollows.

In turn, Theorem 2.7.2 enables us to understand that, as already remarked afterProposition 2.7.1, the non-degenerate continuous covariant representation obtainedin that proposition is actually R-continuous, under the extra condition that allelements of R are non-degenerate. In that case, there is an associated boundedrepresentation of the crossed product again, and the following result, in which someother main results of this section have been included again for future reference, showsadditionally that this two-step process is the identity.

Theorem 2.7.3. Let (A,G, α) be a Banach algebra dynamical system, where A hasa bounded approximate left identity, and let R be a non-empty uniformly boundedclass of non-degenerate continuous covariant representations. Let (iRA , i

RG) be the

non-degenerate R-continuous covariant representation of (A,G, α) on (A oα G)R,as in Proposition 2.6.4 and Theorem 2.7.2.

Suppose that T is a non-degenerate bounded representation of (A oα G)R in aBanach space X, and let T be the associated representation ofMl((AoαG)R) in X,as in Theorem 2.6.1. Then the pair (T iRA , T iRG) is a non-degenerate R-continuouscovariant representation of (A,G, α) in X, and the corresponding non-degenerate

bounded representation((T iRA ) o (T iRG)

)Rof (AoαG)R in X coincides with T .

In particular,∥∥(T iRA ) o (T iRG)

∥∥R = ‖T‖.If a closed subspace of X is invariant for T , it is invariant for T iA and

T iG, and if Y is a Banach space, S : (A oα G)R → B(Y ) a representationand Φ ∈ B(X,Y ) intertwines T and S, then Φ intertwines (T iRA , T iRG) and(S iRA , S iRG).

If, in addition, (A,G, α), R, and T are involutive, then (T iRA , T iRG) is invo-lutive.

Proof. The statements concerning invariant subspaces, intertwiners and involutionshave already been proven in Proposition 2.7.1.

Denote π := T iRA and U := T iRG . Proposition 2.7.1 asserts that (π, U) is a con-tinuous covariant representation, hence its integrated form πoU : Cc(G,A)→ B(X)is defined. We claim that, for all f ∈ Cc(G,A),

π o U(f) = T (qR(f)). (2.7.4)


Assuming this for the moment, we see that, for all f ∈ Cc(G,A),

‖π o U(f)‖ =∥∥T (qR(f))

∥∥ ≤ ‖T‖∥∥qR(f)∥∥R = ‖T‖σR(f).

Hence (π, U) is R-continuous, and consequently the corresponding bounded repre-sentation (π o U)R : (A oα G)R → B(X) can indeed be defined and we conclude,using the definition and (2.7.4), that (π o U)R(qR(f)) = π o U(f) = T (qR(f)),for all f ∈ Cc(G,A). By the density of qR(Cc(G,A)) in (A oα G)R, this impliesthat (π o U)R = T . The statement concerning the norms then follows from Theo-rem 2.5.6.

Hence it remains to establish (2.7.4). For this, let f, g ∈ Cc(G,A). Then (2.6.2)implies that

Us T (qR(g)) = T (iRG(s)) T (qR(g))

= T (iRG(s)(qR(g)))

= T(qR(iG(s)g)

).

Similarly, we have, for all a ∈ A, and h ∈ Cc(G,A),

π(a) T (qR(h)) = T(qR(iA(a)h)

)x.

Combining these, we find that, for s ∈ G,

π(f(s))Us T (qR(g)) = π(f(s)) T(qR(iG(s)g)

)= T

(qR(iA(f(s))iG(s)g)

)= T

(iRA (f(s))iRG(s)qR(g)

).

We conclude that, for all f, g ∈ Cc(G,A),

π o U(f) T (qR(g)) =

∫G

T(iRA (f(s))iRG(s)qR(g)

)ds. (2.7.5)

On the other hand, with λ denoting the left regular representation of (A oα G)R,Theorem 2.7.2 implies that, for f, g ∈ Cc(G,A),

qR(f) ∗ qR(g) = λ(qR(f))qR(g)

= (iRA o iRG)R(qR(f))qR(g)

= iRA o iRG(f)qR(g)

=

∫G

iRA (f(s))iRG(s)qR(g) ds.

Applying the bounded homomorphism T to this relation yields

T (qR(f)) T (qR(g)) =

∫G

T(iRA (f(s))iRG(s)qR(g)

)ds, (2.7.6)


for all f, g ∈ Cc(G,A). For x ∈ X, comparing (2.7.5) and (2.7.6) and applying themto x, we see that

π o U(f)(T (qR(g))x

)= T (qR(f))

(T (qR(g))x

). (2.7.7)

Now since T is a non-degenerate bounded representation of (AoαG)R, the restrictionof T to the dense subalgebra qR(Cc(G,A) must be non-degenerate as well, andso elements of the form T (qR(g))x are dense in X. Hence (2.7.7) implies thatπ o U(f) = T (qR(f)) holds for all f ∈ Cc(G,A), as desired.

2.8 Representations: general correspondence

In this section, which can be viewed as the conclusion of the analysis in the precedingparts of this paper, we put the pieces together without too much extra effort. We givereferences to the relevant definitions, in order to enhance accessibility of the resultsto the reader who is not familiar with the details of the Sections 2.2 through 2.7.Section 2.9 contains some applications.

As an introductory remark for the reader who is familiar with the precedingsections, we note that Theorem 2.5.6 describes the properties of the passage fromR-continuous covariant representations of (A,G, α) to bounded representations of(A oα G)R. Such a passage is always possible, without further assumptions onthe Banach algebra dynamical system or the covariant representations. Proposi-tion 2.7.1, valid under the condition that A has a bounded approximate left identity,goes in the opposite direction, but it is only for non-degenerate bounded representa-tions of (A oα G)R that a (non-degenerate) continuous covariant representation of(A,G, α) is constructed. If one starts with an non-degenerate R-continuous covari-ant representation of (A,G, α), passes to the associated non-degenerated boundedrepresentation of (AoαG)R, and then goes back to (A,G, α) again, the same Propo-sition 2.7.1 shows that one retrieves the original covariant representation of (A,G, α).If, in addition, all elements of R are themselves non-degenerate, then Proposi-tion 2.7.1 can be improved to Theorem 2.7.3, where it is concluded that the (non-degenerate) continuous covariant representation of (A,G, α) as constructed froma non-degenerate bounded representation of (A oα G)R is actually R-continuous.Hence it is possible to go in the first direction again, thus obtaining a boundedrepresentation of (A oα G)R, and, according to the same Theorem 2.7.3, this isthe representation of (A oα G)R one started with. As it turns out, if we imposethese conditions on A (having a bounded approximate left identity) and R (con-sisting of non-degenerate continuous covariant representations), and restrict our at-tention to non-degenerate R-continuous covariant representations of (A,G, α) andnon-degenerate bounded representations of (Aoα G)R, then we obtain a bijection,according to our main general result, the general correspondence in Theorem 2.8.1below.

We now turn to the formulation of the result, recalling the relevant notions anddefinitions as a preparation, and introducing two new notations for (covariant) rep-resentations of a certain type. If (A,G, α) is a Banach algebra dynamical system

2.8. REPRESENTATIONS: GENERAL CORRESPONDENCE 67

(Definition 2.2.10), R is a non-empty uniformly bounded (Definition 2.3.1) class ofnon-degenerate continuous covariant representations (Definition 2.2.12) of (A,G, α),and X is a non-empty class of Banach spaces, we let CovrepRnd,c((A,G, α),X ) de-note the non-degenerate R-continuous (Definitions 2.3.2 and 2.5.1) representationsof (A,G, α) in spaces from X , and we let Repnd,b((A oα G)R,X ) denote the non-

degenerate bounded representations of the crossed product (A oα G)R (Defini-tion 2.3.2) in spaces from X . There need not be a relation between the representationspaces corresponding to the elements of R and the spaces in X .

Furthermore, we let IR denote the assignment (π, U) → (π o U)R, sending anR-continuous covariant representation of (A,G, α) to a bounded representation of(A oα G)R, as explained following Remark 2.5.2. If A has a bounded approximateleft identity, then we let SR denote the assignment T → (T iRA , T iRG), as in Propo-sition 2.7.1 and Theorem 2.7.3, sending a non-degenerate bounded representation of(A oα G)R to a non-degenerate continuous covariant representation of (A,G, α),obtained by first constructing a non-degenerate bounded representation T of theleft centralizer algebra of (AoαG)R, compatible with T , and subsequently compos-ing this with the canonical continuous covariant representation (iRA , i

RA ) of (A,G, α)

in (A oα G)R, the images of which actually lie in this left centralizer algebra (seeProposition 2.6.4).

The notations IR and SR are meant to suggest “integration” and “separation”,respectively.

Finally, we recall the notions of an involutive Banach algebra dynamical sys-tem (Definition 2.2.10), of an involutive representation of such a system (Defini-tion 2.2.12), and of bounded intertwining operators between (covariant) representa-tions (final part of Section 2.2.3).

Theorem 2.8.1 (General correspondence theorem). Let (A,G, α) be a Banach al-gebra dynamical system, where A has a bounded approximate left identity, let R bea non-empty uniformly bounded class of non-degenerate continuous covariant repre-sentations of (A,G, α), and let X be a non-empty class of Banach spaces. Then therestriction of IR yields a bijection

IR : CovrepRnd,c((A,G, α),X )→ Repnd,b((Aoα G)R,X ),

and the restriction of SR yields a bijection

SR : Repnd,b((Aoα G)R,X )→ CovrepRnd,c((A,G, α),X ),

In fact, these restrictions of IR and SR are inverse to each other.Furthermore, both these restrictions of IR and SR preserve the set of closed

invariant subspaces for an element of their domain, as well as the Banach space ofbounded intertwining operators between two elements of their domain.

If (A,G, α) and R are involutive, then both these restrictions of IR and SRpreserve the property of being involutive.

Proof. According to Theorem 2.5.6 and Theorem 2.7.3, and also taking into accountthat IR and SR obviously preserve the representation space, these restricted maps


are indeed meaningfully defined with domains and codomains as in the statement.Proposition 2.7.1 shows that SR(IR((π, U))) = (π, U), for each non-degenerateR-continuous covariant representation (π, U) of (A,G, α), whereas Theorem 2.7.3asserts that IR(SR(T )) = T , for each non-degenerate bounded representation T of(Aoα G)R. This settles the bijectivity statements.

Theorem 2.5.6 and Theorem 2.7.3 contain the statements about preservation ofclosed invariant subspaces, intertwining operators and the property of being involu-tive.

Remark 2.8.2. The map SR, associated with a Banach algebra dynamical sys-tem (A,G, α), where A has a bounded left approximate identity, and a non-emptyuniformly bounded class R of non-degenerate continuous covariant representationsof (A,G, α), can be made explicit by recalling how Theorem 2.6.1 was used in itsdefinition. Indeed, let T be a non-degenerate bounded representation of (Aoα G)R

in a Banach space X. We recall the bounded approximate left identity (qR(zV ⊗ui))of (A oα G)R of Theorem 2.4.4; here (ui) is a bounded approximate left identityof A, V runs through a neighbourhood basis Z of e ∈ G, of which all elements arecontained in a fixed compact subset of G, and zV ∈ Cc(G) is positive, with totalintegral equal to 1, and supported in V . Let x ∈ X. Then by (2.6.3) we find, fora ∈ A,

(T iRA )(a)x = T (iRA (a))x (2.8.1)

= lim(V,i)

T[iRA (a)

(qR(zV ⊗ ui)

)]x

= lim(V,i)

T[qR(zV ⊗ aui)

]x,

and, for r ∈ G,

(T iRG)(r)x = T (iRG(r))x (2.8.2)

= lim(V,i)

T[iRG(r)

(qR(zV ⊗ ui)

)]x

= lim(V,i)

T[qR(zV (r−1·)⊗ αr(ui))

]x.

Denoting s-lim for the limit in the strong operator topology, it follows that

SR(T ) =

(a 7→ s- lim

(V,i)T[qR(zV ⊗ aui)

], r 7→ s- lim

(V,i)T[qR(zV (r−1·)⊗ αr(ui))

]).

Remark 2.8.3. Any non-degenerate continuous covariant representation (π, U)of (A,G, α) in a Banach space X is an element of CovrepRnd,c((A,G, α),X ) withR = (π, U) and X = X. If A has a bounded approximate left identity, then, fora ∈ A, r ∈ G and x ∈ X, inserting (2.8.1) and (2.8.2) into SR(IR((π, U))) = (π, U),

2.8. REPRESENTATIONS: GENERAL CORRESPONDENCE 69

as known from Theorem 2.8.1, yields, with the zV ⊗ ui as in Remark 2.8.2,

π(a)x =(

(π o U)R iRA)

(a)x

= lim(V,i)

(π o U)R[qR(zV ⊗ aui)

]x

= lim(V,i)

∫G

zV (s)π(aui)Usx ds,

Urx =(

(π o U)R iRG)

(r)x

= lim(V,i)

(π o U)R[qR(zV (r−1·)⊗ αr(ui))

]x

= lim(V,i)

∫G

zV (r−1s)π(αr(ui))Usx ds.

These formulas, valid for an arbitrary non-degenerate continuous covariant rep-resentation (π, U) of (A,G, α), where A has a bounded approximate left identity,can also be obtained more directly, by writing x = lim(V,i) π o U(zV ⊗ ui)x (usingRemark 2.2.8) and then using (2.5.3) in Proposition 2.5.3.

Remark 2.8.4. One also has norm estimates related to the maps IR and SR.As to IR, if we assume that (A,G, α) is a Banach algebra dynamical system,

and that R is a non-empty uniformly bounded class of continuous covariant repre-sentations of (A,G, α), then, if (π, U) is an R-continuous covariant representationof (A,G, α), (2.3.3) yields that∥∥IR((π, U))qR(f)

∥∥ = ‖π o U(f)‖ ≤ ‖π‖ ‖f‖L1(G,A) sups∈supp (f)

‖Us‖ , (2.8.3)

for all f ∈ Cc(G,A).In order to give estimates for SR, we assume that (A,G, α) is a Banach algebra

dynamical system, with A having an approximate left identity, and that R is a non-empty uniformly bounded class of continuous covariant representations of (A,G, α).We recall the relevant constants

CR = sup(π,U)∈R

‖π‖ , νR(r) = sup(π,U)∈R

‖Ur‖ , NR = infV ∈Z

supr∈V

νR(r),

where Z is a neighbourhood basis of e ∈ G which is contained in a fixed compactset (see Definition 2.4.3 and the subsequent paragraph, showing that NR does notdepend on the choice of such Z). Furthermore, if A is a normed algebra with abounded approximate left identity, then we recall that MAl denotes the infimum ofthe upper bounds of the approximate left identities of A, and that we write MRl for

M(AoαG)R

l . Then, if T is a non-degenerate bounded representation of (A oα G)R,Proposition 2.7.1 shows that, for a ∈ A,∥∥(T iRA) (a)

∥∥ ≤MRl ‖T‖ sup(π,U)∈R

‖π(a)‖ ≤MRl ‖T‖CR ‖a‖ .


In particular, ∥∥T iRA∥∥ ≤MRl CR ‖T‖ . (2.8.4)

Proposition 2.7.1 also yields that, for r ∈ G,∥∥(T iRG) (r)∥∥ ≤MRl ‖T‖ νR(r). (2.8.5)

Furthermore, by Corollary 2.4.6,

MRl ≤ CRMAl N

R. (2.8.6)

2.9 Representations: special correspondences

In this section, we discuss some special cases of the crossed product construction,based on Theorem 2.8.1. In the first part, we are concerned with a general algebraand group, and make the correspondence between (covariant) representations moreexplicit in a number of cases. In the second part, we consider Banach algebra dy-namical systems where the algebra is trivial. This leads, amongst others, to whatcould be called group Banach algebras associated with a class of Banach spaces.The third part covers the case of a trivial group. Here the machinery as developedin the previous sections is not necessary, and Theorem 2.8.1, although applicable,does, in fact, not yield optimal results. In this case the crossed product is merelythe completion of a quotient of the algebra, and the correspondence between rep-resentations is then standard, but we have nevertheless included the results for thesake of completeness of the presentation.

2.9.1 General algebra and group

Theorem 2.8.1 gives, for each class X of Banach spaces, a bijection between non-degenerate R-continuous covariant representations of (A,G, α) and non-degeneratebounded representations of (AoαG)R in spaces from X . By definition, a continuouscovariant representation (π, U) is R-continuous if there exists a constant C such that‖π o U(f)‖ ≤ C sup(ρ,V )∈R ‖ρo V (f)‖, for all f ∈ Cc(G,A). One would like tomake this condition more explicit in terms of ‖π‖ and ‖Ur‖, for r ∈ G. For certainsituations, this is indeed feasible (possibly by also restricting the maps IR and SRin Theorem 2.8.1 to suitable subsets of their domains) on basis of the estimates inRemark 2.8.4. Our basic theorem in this vein is the following.

Theorem 2.9.1. Let (A,G, α) be a Banach algebra dynamical system, where, foreach ε > 0, A has a (1+ε)-bounded approximate left identity. Let Z be a neighbour-hood basis of e ∈ G contained in a fixed compact set, let ν : G → [0,∞) be boundedon compact sets and satisfy infV ∈Z supr∈V ν(r) = 1. Let R be a non-empty classof non-degenerate continuous covariant representations of (A,G, α), such that, for(π, U) ∈ R, π is contractive and ‖Ur‖ ≤ ν(r), for all r ∈ G.

Let X be a class of Banach spaces, and suppose that R contains the class R′, con-sisting of all non-degenerate continuous covariant representations (π, U) of (A,G, α)in spaces from X , where π is contractive and ‖Ur‖ ≤ ν(r), for all r ∈ G.

2.9. REPRESENTATIONS: SPECIAL CORRESPONDENCES 71

If R′ is non-empty, then the map (π, U) 7→ (π o U)R is a bijection between R′and the non-degenerate contractive representations of (Aoα G)R in spaces from X .This map preserves the set of closed invariant subspaces, as well as the Banach spaceof bounded intertwining operators between two elements of R′.

If (A,G, α) and R are involutive, then this bijection preserves the property ofbeing involutive.

Proof. We use Theorem 2.8.1. Suppose that (π, U) ∈ R′ ⊂ R, then certainly(π, U) is R-continuous, so that R′ ⊂ CovrepRnd,c((A,G, α),X ). Hence the results

of that theorem are applicable, and we must show that IR and SR are bijectionsbetween R′ and the non-degenerate contractive representations of (A oα G)R inthe elements of X , which form a subset of Repnd,b((A oα G)R,X ). Suppose that

(π, U) ∈ R′ ⊂ R, then IR((π, U)) = (π o U)R is obviously contractive by the verydefinition of σR and (AoαG)R in Definition 2.3.2. Conversely, suppose that T is anon-degenerate contractive representation of (Aoα G)R in a space from X . In thenotation of Remark 2.8.4, we have CR ≤ 1, and MA

l ≤ 1. By definition of νR wehave νR ≤ ν, so the condition on ν implies that NR ≤ 1. From (2.8.6), we thenconclude that MRl ≤ 1. Thus SR(T ) = (T iRA , T iRG) is not only known to bea non-degenerate covariant representation of (A,G, α) in a space from X by Theo-rem 2.8.1, but in addition we know from (2.8.4) that

∥∥T iRA∥∥ ≤ 1, and from (2.8.5)

that∥∥(T iRG)(r)

∥∥ ≤ νR(r) ≤ ν(r), for all r ∈ G, so SR(T ) ∈ R′. This settles themain part of the present theorem, and the rest is immediate from Theorem 2.8.1.

As a particular case of Theorem 2.9.1, we letR andR′ coincide, and we specializeto ν ≡ 1. Note that this condition ‖Ur‖ ≤ 1, for all r ∈ G, is equivalent to requiringthat U is isometric. Thus we obtain the following.

Theorem 2.9.2. Let (A,G, α) be a Banach algebra dynamical system, where, foreach ε > 0, A has a (1 + ε)-bounded approximate left identity. Let X be a class ofBanach spaces, and let R consist of all non-degenerate continuous covariant repre-sentations (π, U) of (A,G, α) in spaces from X , such that π is contractive and Ur isan isometry, for all r ∈ G.

If R is non-empty, then the map (π, U) 7→ (πoU)R is a bijection between R andthe non-degenerate contractive representations of (AoαG)R in spaces from X . Thismap preserves the set of closed invariant subspaces, as well as the Banach space ofbounded intertwining operators between two elements of R.


Specializing Theorem 2.9.2 in turn to the involutive case yields the following.We recall from the third part of Remark 2.3.3 that (A oα G)R is a C∗-algebra if(A,G, α) and R are involutive.

Theorem 2.9.3. Let (A,G, α) be a Banach algebra dynamical system, where A isa Banach algebra with bounded involution and where G acts as involutive automor-phisms on A. Assume that, for each ε > 0, A has a (1+ε)-bounded approximate left


identity. Let H be a class of Hilbert spaces, and let R consist of all non-degeneratecontinuous covariant representations (π, U) of (A,G, α) in elements of H, such thatπ is contractive and involutive, and Ur is unitary, for all r ∈ G.

If R is non-empty, then the map (π, U) 7→ (π o U)R is a bijection between Rand the non-degenerate involutive representations of the C∗-algebra (A oα G)R inspaces from H. This map preserves the set of closed invariant subspaces, as well asthe Banach space of bounded intertwining operators between two elements of R.

Remark 2.9.4. Note that Theorem 2.9.3 applies to all C∗-dynamical systems, sincethen A has a 1-bounded approximate left identity. In that case, if R is non-empty,then (A oα G)R can be considered as the C∗-crossed product associated with theC∗-dynamical system (A,G, α) and the Hilbert spaces from H. If H consists of allHilbert spaces, then the associated C∗-algebra (A oα G)R is commonly known asthe crossed product AoαG, as in [51]. Surely R is then non-empty, since it containsthe zero representation on the zero space. However, more is true: the Gelfand-Naimark theorem furnishes a faithful non-degenerate involutive representation ofA in a Hilbert space, and then [51, Lemma 2.26] provides a covariant involutiverepresentation of (A,G, α) (which is non-degenerate by [51, Lemma 2.17]), of whichthe integrated form is a faithful representation of Cc(G,A). As a consequence, σR

is then an algebra norm on Cc(G,A), rather than a seminorm, and the quotientconstruction as in the present paper for the general case is then not necessary.

We conclude this section with a preparation for the sequel [22], where we willshow that under certain conditions (A oα G)R is (isometrically) isomorphic to theBanach algebra L1(G,A) with a twisted convolution product. With this in place,we will then also be able to show how well-known results about (bi)-modules forL1(G) ([20, Assertion VI.1.32], [24, Proposition 2.1]) fit into the general frameworkof crossed products of Banach algebras.

The preparatory result we will then require is the following; the function νR

figuring in it is defined in Remark 2.8.4.

Theorem 2.9.5. Let (A,G, α) be a Banach algebra dynamical system, where Ahas a bounded approximate left identity. Let D ≥ 0, and let R be a non-emptyclass of non-degenerate continuous covariant representations (π, U) of (A,G, α),such that νR(r) ≤ D, for all r ∈ G. Assume that there exists C1 ≥ 0 such that‖f‖L1(G,A) ≤ C1 sup(π,U)∈R ‖π o U(f)‖, for all f ∈ Cc(G,A).

Let X be a class of Banach spaces. Then the map (π, U) 7→ (π o U)R is a bijec-tion between the non-degenerate continuous covariant representations of (A,G, α) inspaces from X for which there exists a constant CU , such that ‖Ur‖ ≤ CU , for allr ∈ G, and the non-degenerate bounded representations of (AoαG)R in spaces fromX . This map preserves the set of closed invariant subspaces, as well as the Banachspace of bounded intertwining operators between two elements of R.


Proof. We apply Theorem 2.8.1 and show that IR and SR map the sets of (covari-ant) representations as described in the present theorem into each other. Suppose


(π, U) is a non-degenerate continuous covariant representations of (A,G, α) in aspace from X for which U is uniformly bounded. Then by (2.3.3), the assumptionimplies that, for all f ∈ Cc(G,A),

‖π o U(f)‖ ≤ ‖π‖D ‖f‖L1(G,A) ≤ ‖π‖DC1σR(f),

and so π o U is R-continuous and hence induces a non-degenerate bounded repre-sentation (π o U)R of (Aoα G)R.

Conversely, let T be a non-degenerate bounded representation of (A oα G)R ina space from X . Then (T iRA , T iRG) is not only known to be a non-degeneratecontinuous covariant representation (π, U) of (A,G, α), by Theorem 2.8.1, but inaddition (2.8.5), together with νR(r) ≤ D for all r ∈ G, shows that T iG isuniformly bounded.

Remark 2.9.6. Under the hypotheses of Theorem 2.9.5, (2.3.3) implies that we alsohave σR(f) ≤ CRD ‖f‖L1(G,A), so that σR and ‖ . ‖L1(G,A) are equivalent algebra

norms on Cc(G,A). As a consequence, (AoαG)R and L1(G,A) are isomorphic Ba-nach algebras, and the non-degenerate continuous covariant representations (π, U) inspaces from X , as described in the theorem, are in bijection with the non-degeneratebounded representations of L1(G,A) in the elements of X . The questions when thecondition ‖f‖L1(G,A) ≤ C1σ

R(f) is actually satisfied, and when (A oα G)R and

L1(G,A) are even isometrically isomorphic Banach algebras, will be tackled in thesequel ([22]), to which we also postpone further discussion.

2.9.2 Trivial algebra: group Banach algebras

We now specialize the results of Theorem 2.9.1 to the case where the algebra isequal to the field K, and the group acts trivially on it. We start by making somepreliminary remarks.

The general representation of K in a Banach space X is given by letting λ ∈ K actas λP , where P ∈ B(X) is an idempotent. Therefore, the only non-degenerate rep-resentation of K in X is the canonical one, canX : K→ B(X), obtained for P = idX .As a consequence, the non-degenerate covariant representations of (K, G, triv) in agiven Banach space X are in bijection with the strongly continuous representationsof G in that Banach space, by letting (canX , U) correspond to U . Likewise, the non-degenerate involutive continuous covariant representations of (K, G, triv) in a givenHilbert space are in bijection with the unitary strongly continuous representationsof G in that Hilbert space.

The Banach algebra dynamical system (K, G, triv) has a non-degenerate con-tinuous covariant representation in each Banach space X, with G acting as isome-tries, namely, by letting the field act as scalars and letting the group act trivially.Likewise, there is a non-degenerate involutive continuous covariant representation of(K, G, triv) in each Hilbert space, with G acting as unitaries. Therefore, the hypoth-esis in the theorems in Section 2.9.1 that certain classes of non-degenerate continuouscovariant representations are non-empty is sometimes redundant. Furthermore, the


hypothesis on the existence of a suitable bounded approximate left identity in K isobviously always satisfied.

We introduce some shorthand notation. If R is a class of strongly continuousrepresentations of G, then R := (canXU , U) : U ∈ R is a uniformly boundedclass of continuous covariant representations of (K, G, triv) precisely if there existsa function ν : G → [0,∞), which is bounded on compact subsets of G, and suchthat ‖Ur‖ ≤ ν(r), for all U ∈ R, and all r ∈ G. In that case, the associated crossed

product (K otriv G)R is defined, but we will write (K otriv G)R for short. Thus(Kotriv G)R is obtained by starting with Cc(G) in its usual convolution structure,introducing the seminorm

σR(f) = supU∈R

∥∥∥∥∫G

f(s)Us ds

∥∥∥∥ (f ∈ Cc(G)),

and completing Cc(G)/ker(σR) in the norm induced on this quotient by σR. Asbefore, we let qR denote the canonical map from Cc(G) into (K otriv G)R. If Uis a strongly continuous representation of G, then we let U(f) =

∫Gf(s)Us ds,

which corresponds to canXU oU(f) in the previous sections. Then U will be calledR-continuous if there exists a constant C such that ‖U(f)‖ ≤ CσR(f), for all

f ∈ Cc(G); this corresponds to canXU oU being R-continuous. In that case, there isan associated bounded representation of (K otriv G)R, denoted by UR rather than(canXU o U)R, which is given on the dense subalgebra qR(Cc(G)) of (K otriv G)R

by

UR(qR(f)) =

∫G

f(s)Us ds (f ∈ Cc(G)).

With these notations, Theorems 2.9.1 specializes to the following result.

Theorem 2.9.7. Let G be a locally compact group. Let Z be a neighbourhoodbasis of e ∈ G contained in a fixed compact set, let ν : G → [0,∞) be bounded oncompact sets and satisfy infV ∈Z supr∈V ν(r) = 1. Let R be a non-empty class ofstrongly continuous representations of G on Banach spaces, such that, for U ∈ R,‖Ur‖ ≤ ν(r), for all r ∈ G.

Let X be a class of Banach spaces, and suppose that R contains the class R′,consisting of all strongly continuous representations U of G in spaces from X , suchthat ‖Ur‖ ≤ ν(r), for all r ∈ G.

Then the map which sends U ∈ R′ to UR is a bijection between R′ and the non-degenerate contractive representations of (K otriv G)R in the Banach spaces fromX . This map preserves the set of closed invariant subspaces, as well as the Banachspace of bounded intertwining operators between two elements of R′.

If all elements from R are unitary strongly continuous representations, then thisbijection lets unitary strongly continuous representations of G in R′ correspond toinvolutive representations of the C∗-algebra (Cotriv G)R.

Specializing the above result to R = R′ and ν ≡ 1 (or Theorem 2.9.2 to the caseof the trivial algebra), we obtain the following.


Theorem 2.9.8. Let G be a locally compact group. Let X be a non-empty class ofBanach spaces, and let R be the class of all isometric strongly continuous representa-tions of G in spaces from X . Then the map which sends U ∈ R to UR is a bijectionbetween R and the non-degenerate contractive representations of the Banach algebra(K otriv G)R in the Banach spaces from X . This map preserves the set of closedinvariant subspaces, as well as the Banach space of bounded intertwining operators.

The following is a consequence of Theorem 2.9.3.

Theorem 2.9.9. Let G be a locally compact group, let H be a non-empty class ofHilbert spaces, and let R consists of all unitary strongly continuous representationsof G in the Hilbert spaces from H.

Then the map which sends U ∈ R to UR is a bijection between R and the non-degenerate involutive representations of the C∗-algebra (K otriv G)R in the Hilbertspaces from H. This map preserves the set of closed invariant subspaces, as well asthe Banach space of bounded intertwining operators between two elements of R.

Remark 2.9.10. The Banach algebra in Theorem 2.9.8 could be called the groupBanach algebra BX (G) of G associated with the (isometric strongly continuous rep-resentations of G in the) Banach spaces from X . As explained in the Introduction,these algebras, and their possible future role in decomposition theory for grouprepresentations, were part of the motivation underlying the present paper. The C∗-algebra in Theorem 2.9.9 is the group Banach algebra BH(G), which in this casehas additional structure as a C∗-algebra. If H consists of all Hilbert spaces, thenthe group Banach algebra BH(G) is, of course, what is commonly known as C∗(G),“the” group C∗-algebra of G.

2.9.3 Trivial group: enveloping algebras

We conclude with a few remarks on the case where the group is equal to the trivialgroup, e, acting trivially on A. In this situation Cc(e, A) ∼= A as abstractalgebras, so, if R is a class of representations of A in Banach spaces, for which thereexists a constant C, such that ‖π‖ ≤ C, for all π ∈ R, then one naturally associatesa uniformly bounded class of continuous covariant representations of (A, e, triv)with R, and, with obvious notational convention, constructs the crossed product(Aotriv e)R. This crossed product is simply the completion of A/ ker(σR) in thenorm corresponding to the seminorm σR on A, defined by σR(a) = supπ∈R ‖π(a)‖,for a ∈ A.

In principle, one could apply Theorem 2.9.1 in this situation, but then one wouldneed to require A to have a bounded left approximate identity. The reason underly-ing this is that, in general, there are no homomorphisms of the algebra or the groupinto the crossed product, so that the most natural idea to obtain representations ofalgebra and group from a representation of the crossed product, namely, to composea given representation of the crossed product with such homomorphisms, will notwork in general. In our approach in previous sections, the left centralizer algebraof the crossed product, into which the algebra and group do map, provided an al-ternative, but then one needs a bounded left approximate identity of the algebra,


in order to be able to construct a representation of the left centralizer algebra froma (non-degenerate) representation of the crossed product. In the present case of atrivial group, however, one needs only a homomorphism of the algebra A into thecrossed product, and since this is a completion of A/ ker(σR), this clearly exists andthe machinery we had to employ in previous sections is now not required. Also,the non-degeneracy of representations (needed to construct representations of theleft centralizer algebra) is no longer an issue. One simply applies Lemma 2.2.20,and thus obtains the following elementary and well-known theorem for the crossedproduct (Aotriv e)R, which we include for the sake of completeness.

Theorem 2.9.11. Let A be a Banach algebra, and let R be a non-empty uniformlybounded class of representations of A in Banach spaces. Let σR(a) = supπ∈R ‖π(a)‖,for a ∈ A, denoted the associated seminorm, and let AR be the completion ofA/ ker(σR) in the norm corresponding to σR on A.

Let X be a non-empty class of Banach spaces, and say that a bounded represen-tation π of A in a space from X is R-continuous if there exists a constant C, suchthat ‖π(a)‖ ≤ CσR(a), for all a ∈ A. In that case, define the norm of π as theminimal such C.

Then the R-continuous representations of A in the spaces from X correspondnaturally with the bounded representations of AR in spaces from X . This corre-spondence preserves the norms of the representations, the set of closed invariantsubspaces, and the Banach spaces of bounded intertwining operator.

If A has a (possibly unbounded) involution, and if all elements of R are involutivebounded representations, then this natural correspondence sets up a bijection betweenthe R-continuous involutive bounded representations of A in the Hilbert spaces in X ,and the involutive representations of the C∗-algebra AR in those spaces.

If A is an involutive Banach algebra with an isometric involution and a boundedapproximate identity, and R consists of all involutive representations in Hilbertspaces, which is uniformly bounded since all such representations are contractiveby [10, Proposition 1.3.7], then the crossed product AR is generally known as theenveloping C∗-algebra of A as described in [10, Section 2.7].

Acknowledgements

The authors thank Miek Messerschmidt for helpful comments and suggestions.

Chapter 3

Positive representations offinite groups in Riesz spaces

This chapter is to appear in International Journal of Mathematics as: M. de Jeuand M. Wortel, “Positive representations of finite groups in Riesz spaces”. It isavailable as arXiv:1109.4505.

Abstract. In this paper, which is part of a study of positive representations oflocally compact groups in Banach lattices, we initiate the theory of positive repre-sentations of finite groups in Riesz spaces. If such a representation has only the zerosubspace and possibly the space itself as invariant principal bands, then the space isArchimedean and finite dimensional. Various notions of irreducibility of a positiverepresentation are introduced and, for a finite group acting positively in a space withsufficiently many projections, these are shown to be equal. We describe the finite di-mensional positive Archimedean representations of a finite group and establish that,up to order equivalence, these are order direct sums, with unique multiplicities, ofthe order indecomposable positive representations naturally associated with transi-tive G-spaces. Character theory is shown to break down for positive representations.Induction and systems of imprimitivity are introduced in an ordered context, wherethe multiplicity formulation of Frobenius reciprocity turns out not to hold.

3.1 Introduction and overview

The theory of unitary group representations is well developed. Apart from its in-trinsic appeal, it has been stimulated in its early days by the wish, originating fromquantum theory, to understand the natural representations of symmetry groups ofphysical systems in L2-spaces. Such symmetry groups do not only yield natural uni-tary representations, but they have natural representations in other Banach spacesas well. For example, the orthogonal group in three dimensions does not only act onthe L2-functions on the sphere or on three dimensional space. It also has a natural

77

78 CHAPTER 3. POSITIVE REPRESENTATIONS OF FINITE GROUPS

isometric action on Lp-functions, for all p, and this action is strongly continuousfor finite p. Moreover, this action is obviously positive. Thus, for finite p, theseLp-spaces, which are Banach lattices, afford a strongly continuous isometric positiverepresentation of the orthogonal group. It is rather easy to find more examples ofpositive representations: whenever a group acts on a point set, then, more oftenthan not, there is a natural positive action on various naturally associated Banachlattices of functions.

However, in spite of the plenitude of examples of strongly continuous (isometric)positive representations of groups in Banach lattices, the theory of such represen-tations cannot compare to its unitary counterpart. Very little seems to be known.Is there, for example, a decomposition theory into indecomposables for such repre-sentations, as a positive counterpart to that for unitary representations describedin, e.g., [10]? When asking such an—admittedly ambitious—question, it is impor-tant to keep in mind that the unitary theory works particularly well in separableHilbert spaces, i.e., for representations which can all be realized in just one space:`2. Since there is a great diversity of Banach lattices, it is not clear at the time ofwriting whether one can expect a general answer for all these lattices with a degreeof sophistication and uniformity comparable to that for the unitary representationsin this single Hilbert space `2. It may be more feasible, at least for the moment,to aim at a better understanding of positive representations on specific classes ofBanach lattices. For example, the results in [23] show that, in the context of a Pol-ish group acting on a Polish space with an invariant measure, it is indeed possibleto decompose—in terms of Banach bundles rather than in terms of direct integralsas in the unitary case—the corresponding isometric positive representation in Lp-spaces (1 ≤ p <∞) into indecomposable isometric positive representations. To ourknowledge, this is the only available decomposition result at this time. Since thisresult covers only representations originating from an action on the underlying pointset, we still cannot decide whether a general (isometric) positive representation ofa (Polish) group in such Banach lattices can be decomposed into indecomposablepositive representations, and more research is necessary to decide this. This, how-ever, is already a relatively advanced issue: as will become clear below, it is easy toask very natural basic questions about positive representations of locally compactgroups in Banach lattices which need answering. This paper, then, is a contributionto the theory of such representations, with a hoped-for decomposition theory intoindecomposable positive representations in mind as a leading and focusing theme,and starting with the obviously easiest case: the finite groups. We will now globallydiscuss it contents.

If a group G acts as positive operators on a Banach lattice E, then the naturalquestion is to ask whether it is possible to decompose E = L⊕M as a G-invariantorder direct sum. In that case, L and M are automatically projection bands andeach other’s disjoint complement. Since bands in a Banach lattice are closed, the de-composition is then automatically also topological, and the original representationssplits as an order direct sum of the positive subrepresentations on the Banach latticesL and M . If such a decomposition is only trivially possible, then we will call therepresentation (order) indecomposable, a terminology already tacitly used above. Is

3.1. INTRODUCTION AND OVERVIEW 79

it then true that every indecomposable positive representation of a finite group Gin a Banach lattice is finite dimensional, as it is for unitary representations? This isnot the case: the trivial group acting on C([0, 1]), which has only trivial projectionbands, provides a counterexample. Is it then perhaps true when we narrow downthe class of spaces to better behaved ones, and ask the same question for Banachlattices with the projection property? After all, since bands are now complementedby their disjoint complement, they seem close to Hilbert spaces where the proof forthe corresponding statement in the unitary case is a triviality, and based on thiscomplementation property. Indeed, if x 6= 0 is an element of the Hilbert space underconsideration where the finite group G acts unitarily and irreducibly, then the orbitG · x spans a finite dimensional, hence closed, nonzero subspace which is clearly in-variant and invariantly complemented. Hence the orbit spans the space, which mustbe finite dimensional. In an ordered context this proof breaks down. Surely, one canconsider the band generated by the orbit of a nonzero element, which is invariantlycomplemented in an order direct sum if the space is assumed to have the projectionproperty. Hence this band is equal to the space, but since there is no guarantee thatit is finite dimensional, once cannot reach the desired conclusion along these lines.We have not been able to find an answer to this finite dimensionality question forBanach lattices with the projection property in the literature, nor could a numberof experts in positivity we consulted provide an answer. The best available resultin this vein seems to be [43, Theorem III.10.4], which implies as a special case thata positive representation of a finite group in a Banach lattice with only trivial in-variant closed ideals is finite dimensional. Still, this does not answer our questionconcerning the finite dimensionality of indecomposable positive representations of afinite group in a Banach lattice with the projection property. The reason is simple:unless one assumes that the lattice has order continuous norm, one cannot concludethat there are only trivial invariant closed ideals from the fact that there are onlytrivial invariant bands. On the other hand: there are no obvious infinite dimensionalcounterexamples in sight, and one might start to suspect that there are none. Thisis in fact the case, and even more holds true: a positive representation of a finitegroup in a Riesz space, with the property that the only invariant principal bands are0 and possibly the space itself, is in a finite dimensional Archimedean space, cf.Theorem 3.3.14 below. As will have become obvious from the previous discussion,such a result is no longer a triviality in an ordered context. It provides an affirma-tive answer to our original question because, for Banach lattices with the projectionproperty, an invariant principal band is an invariant projection band. It also im-plies the aforementioned result that a positive representation of a finite group in aBanach lattice with only trivial invariant closed ideals is finite dimensional. Indeed,an invariant principal band is then an invariant closed ideal.

Thus, even though our original question was in terms of Banach lattices, andmotivated by analytical unitary analogies, an answer can be provided in a moregeneral, topology free context. For finite groups, this is—in fact—perhaps not toobig a surprise. Furthermore, we note that the hypothesis in this finite dimension-ality theorem is not the triviality of all invariant order decompositions, but ratherthe absence of a nontrivial G-invariant object, without any reference to this being


invariantly complemented in an order direct sum or not. It thus becomes clear thatit is worthwhile to not only consider the existence of nontrivial invariant projectionbands (which is the same as the representation being (order) decomposable), but toalso consider the existence of nontrivial invariant ideals, nontrivial invariant bands,etc., for positive representations in arbitrary Riesz spaces, and investigate the inter-relations between the corresponding notions of irreducibility. In the unitary case,indecomposability and irreducibility (for which there is only one reasonable notion)coincide, but in the present ordered context this need not be so. Nevertheless, forfinite groups acting positively in spaces with sufficiently many projections, the mostnatural of these notions of irreducibility are all identical and coincide with (order)indecomposability, cf. Theorem 3.3.16. Again, whereas the corresponding proof forthe unitary case is a triviality, this is not quite so obvious in an ordered context.

As will become apparent in this paper, it is possible to describe all finite di-mensional positive Archimedean representations of a finite group, indecomposableor not. Once this is done, it is not too difficult to show that such finite dimensionalpositive representations can be decomposed uniquely into indecomposable positiverepresentations, and that, up to order equivalence, all such indecomposable positiverepresentations arise from actions of the group on transitive G-spaces, cf. Corol-lary 3.4.11 and Theorem 3.4.10. Since this decomposition into irreducible positiverepresentations with multiplicities is so reminiscent of classical linear representationtheory theory for finite groups, and to Peter-Weyl theory for compact groups, onemight wonder whether parts of character theory also survive. This is hardly the case.For finite groups with only normal subgroups, such as finite abelian groups, there isstill a bijection between characters and order equivalence classes of finite dimensionalindecomposable positive Archimedean representations, cf. Corollary 3.4.14, but weprovide counterexamples to a number of other results as they would be natural toconjecture.

Finally, we consider induction and systems of imprimitivity in an ordered context.As long as topology is not an issue, this can be done from a categorical point ofview for arbitrary groups and arbitrary subgroups. Even though the constructionsare fairly routine, we have included the material, not only as a preparation forfuture more analytical considerations, but also because there are still some differenceswith the linear theory. For example, Frobenius reciprocity no longer holds in itsmultiplicity formulation.

After this global overview we emphasize that, even though this paper containsa basic finite dimensionality result and provides reasonably complete results for fi-nite dimensional positive Archimedean representations of finite groups (in analyticalterms: for positive representations of finite groups in C(K) for K finite), the basicdecomposition issue for positive representations of a finite group in infinite dimen-sional Banach lattices is still open. At the time of writing, the only results in thisdirection seem to be the specialization to finite groups of the results in [23] for Lp-spaces, and of those in [21], which is concerned with Jordan-Holder theory for finitechains of various invariant order structures in Riesz spaces. As long as the group isonly finite, a more comprehensive answer seems desirable.


The structure of this paper is as follows.

In Section 3.2 we introduce the necessary notation and definitions, and we recalla folklore result on transitive G-spaces. Then, in Section 3.3, we investigate the rela-tions between various notions of irreducibility as already mentioned above. We thenestablish one of the main results of this paper, Theorem 3.3.14, stating, amongstothers, that a principal band irreducible positive representation of a finite group isalways finite dimensional. The proof is by induction on the order of the group, anduses a reasonable amount of general basic theory of Riesz spaces. We then continueby examining the structure of finite dimensional positive Archimedean representa-tions of a finite group in Section 3.4. Any such space is lattice isomorphic to Rn, forsome n, and its group Aut+(Rn) of lattice automorphisms is a semidirect product ofSn and the group of multiplication (diagonal) operators, a result which also followsfrom [32, Theorem 3.2.10], but which we prefer to derive by elementary means in ourcontext. Armed with this information we can completely determine the structure ofa positive representations of a finite group in Rn in Theorem 3.4.5: such a positiverepresentation is given by a representation into Sn ⊂ Aut+(Rn), called a permuta-tion representation, and a single multiplication operator. We then determine whentwo positive representations in Rn are order equivalent, which turns out to be thecase precisely when their permutation parts are conjugate. Consequently, in the end,the finite dimensional positive Archimedean representations of a finite group can bedescribed in terms of actions of the group on finite sets. The decomposition resultand the description of indecomposable positive representations already mentionedabove then follow easily. The rest of Section 3.4 is concerned with showing thatcharacter theory does not survive in an ordered context. Finally, in Section 3.5, wedevelop the theory of induction and systems of imprimitivity in the ordered setting,and show that Frobenius reciprocity does not hold in its multiplicity formulation.

3.2 Preliminaries

In this section we will discuss some preliminaries about automorphisms of Rieszspaces, representations, order direct sums of representations and G-spaces. All Rieszspaces in this paper are real. For positive representations in spaces admitting acomplexification it is easy, and left to the reader, to formulate the correspondingcomplex result and derive it from the real case.

Let E be a not necessarily Archimedean Riesz space. If D ⊂ E is any subset,then the band generated by D is denoted by D, and the disjoint complement ofD is denoted by Dd. If T is a lattice automorphism of E, then TD = TD andT (Dd) = (TD)d. The group of lattice automorphisms of E is denoted by Aut+(E).

In this paper Rn is always equipped with the coordinatewise ordering.

Definition 3.2.1. Let G be a group and E a Riesz space. A positive representationof G in E is a group homomorphism ρ : G→ Aut+(E).

For typographical reasons, we will write ρs instead of ρ(s), for s ∈ G.


If (Ei)i∈I is a collection of Riesz spaces, then the order direct sum of this col-lection, denoted ⊕i∈IEi, is the Riesz space with elements (xi)i∈I , where xi ∈ Ei forall i ∈ I, at most finitely many xi are nonzero, and where (xi)i∈I is positive if andonly if xi is positive for all i ∈ I. If additionally ρi : G → Aut+(Ei) is a positiverepresentation for all i ∈ I, then the positive representation

⊕i∈I

ρi : → Aut+

(⊕i∈I

Ei

),

the order direct sum of the ρi, is defined by (⊕i∈Iρi)s := ⊕i∈Iρis, for s ∈ G.

Let ρ : G→ Aut+(E) be a positive representation, and suppose that E = L⊕Mas an order direct sum. Then L and M are automatically projection bands withLd = M by [28, Theorem 24.3]. If both L and M are ρ-invariant, then ρ can beviewed as the order direct sum of ρ acting positively on L and M .

If ρ : G → Aut+(E) and θ : G → Aut+(F ) are positive representations on Rieszspaces E and F , respectively, then a positive map T : E → F is called a positiveintertwiner between ρ and θ if Tρs = θsT for all s ∈ G, and ρ and θ are called orderequivalent if there exists a positive intertwiner between ρ and θ which is a latticeisomorphism.

Turning to the terminology for G-spaces, we let G be a not necessarily finitegroup. A G-space X is a nonempty set X equipped with an action of G; it iscalled transitive if there is only one orbit. For x ∈ X, let Gx denote the subgroups ∈ G : sx = x, the stabilizer of x. If X and Y are G-spaces, then X and Y arecalled isomorphic G-spaces if there is a bijection φ : X → Y , such that sφ(x) = φ(sx)for all s ∈ G and x ∈ X. We let [X] denote the class of all G-spaces isomorphic toX.

If X is a transitive G-space and x ∈ X, then sx 7→ sGx is a G-space isomorphismbetween X and G/Gx with its natural G-action. The next folklore lemma elaborateson this correspondence.

Lemma 3.2.2. Let G be a not necessarily finite group. For each isomorphism class[X] of transitive G-spaces, choose a representative X and x ∈ X. Then the conjugacyclass [Gx] of Gx does not depend on the choices made, and the map [X] 7→ [Gx] is abijection between the isomorphism classes of transitive G-spaces and the conjugacyclasses of subgroups of G.

Proof. It is easy to see that the map is well-defined and surjective. For injectivity,let X and Y be transitive G-spaces, such that [Gx] = [Gy] for some x ∈ X andy ∈ Y . We have to show that [X] = [Y ], or equivalently, G/Gx ∼= G/Gy. Byassumption Gx = rGyr

−1 for some r ∈ G, and the map sGx 7→ sGxr = srGy isthen an isomorphism of G-spaces between G/Gx and G/Gy.

3.3. IRREDUCIBLE AND INDECOMPOSABLE REPRESENTATIONS 83

3.3 Irreducible and indecomposablerepresentations

In the theory of unitary representations of groups, the nonexistence of nontriv-ial closed invariant subspaces is the only reasonable notion of irreducibility of arepresentation, and it coincides with the natural notion of indecomposability of arepresentation. In a purely linear context, irreducibility and indecomposability ofgroup representations need not coincide, however, and the same is true in an or-dered context where, in addition, there are several natural notions of irreducibility.In this section, we are concerned with the relations between the various notions andwe establish a basic finite dimensionality result, Theorem 3.3.14. This is then usedto show that, in fact, the various notions of irreducibility are equivalent for finitegroups if the space has sufficiently many projections, cf. Theorem 3.3.16. We let Gbe a group, to begin with not necessarily finite.

Definition 3.3.1. A positive representation ρ : G → Aut+(E) is called band irre-ducible if a ρ-invariant band equals 0 or E. Projection band irreducibility, idealirreducibility, and principal band irreducibility are defined similarly, as are closedideal irreducibility, etc., in the case of normed Riesz spaces.

Starting our discussion of the implications between the various notions of ir-reducibility, we note that, obviously, band irreducibility implies projection bandirreducibility. If E has the projection property, then the converse holds trivially aswell, since all bands are projection bands by definition, but the next example showsthat this converse fails in general.

Example 3.3.2. Consider the representation of the trivial group on C[0, 1]. Nowevery band is invariant, so this representation is not band irreducible, but C[0, 1]does not have any nontrivial projection bands, and therefore it is projection bandirreducible.

For positive representations on Banach lattices, closed ideal irreducibility impliesband irreducibility. If the Banach lattice has order-continuous norm, then the con-verse holds as well, since then all closed ideals are bands ([32, Corollary 2.4.4]), butonce again this converse fails in general, as the next example shows.

Example 3.3.3. Consider `∞(Z), the space of doubly infinite bounded sequences,and define ρ : Z → Aut+(`∞(Z)) by ρk(xn) := (xn−k), the left regular representa-tion. This representation is not closed ideal irreducible, since the space c0(Z) ofsequences tending to zero is an invariant closed ideal. On the other hand, it is easyto see that any nonzero invariant ideal must contain the order dense subspace offinitely supported sequences, therefore ρ is band irreducible.

Finally, for positive representations on Banach lattices, ideal irreducibility obvi-ously implies closed ideal irreducibility, but again there is an example showing thatthe converse fails in general.


Example 3.3.4. Consider the left regular representation of Z on c0(Z), as in theabove example. Then `1(Z) is an invariant ideal, so this positive representation isnot ideal irreducible, but it is closed ideal irreducible since every nonzero invariantideal must contain the norm dense subspace of finitely supported sequences.

We continue by defining the natural notion of indecomposability for positiverepresentations, which is order indecomposability. As usual, the order direct sumE = L⊕M is called nontrivial if L 6= 0 and L 6= E.

Definition 3.3.5. A positive representation ρ : G→ Aut+(E) is called order inde-composable if there are no nontrivial ρ-invariant order direct sums E = L⊕M .

We will now investigate the conditional equivalence between order indecompos-ability and the various notions of irreducibility.

Lemma 3.3.6. A positive representation ρ : G→ Aut+(E) is order indecomposableif and only if it is projection band irreducible.

Proof. Suppose ρ is order indecomposable, and let B be a ρ-invariant projectionband. We claim that Bd is ρ-invariant. For this, let x ∈ (Bd)+ and s ∈ G. Then(ρsx)∧ y = ρs(x∧ ρ−1

s y) = ρs0 = 0 for all y ∈ B+, so ρsx⊥B, i.e., ρsx ∈ Bd. HenceE = B ⊕Bd is a ρ-invariant order direct sum, so either B = 0 or B = E.

Conversely, suppose that ρ is projection band irreducible. Let E = L⊕M be aρ-invariant order direct sum. Then, as mentioned in the preliminaries, L and M areprojection bands, and therefore L = 0 or M = 0.

Thus order indecomposability is equivalent with projection band irreducibility.We have already seen that the latter property is, in general, not equivalent withband irreducibility, but that equivalence between these two does hold (trivially) ifthe Riesz space has the projection property. However, if the group is finite, we willsee in Lemma 3.3.9 and Theorem 3.3.16 below that these three notion are equivalentunder a much milder assumption on the space, as in the following definition.

Definition 3.3.7. A Riesz space E is said to have sufficiently many projections ifevery nonzero band contains a nonzero projection band.

This notion is intermediate between the principal projection property and theArchimedean property, cf. [28, Theorem 30.4]. In order to show that it is (inparticular) actually weaker than the projection property, which is the relevant featurefor our discussion, we present an example of a Banach lattice which has sufficientlymany projections, but not the projection property.

Example 3.3.8. Let ∆ ⊂ [0, 1] be the Cantor set, and let E = C(∆). Then [32,Corollary 2.1.10] shows that bands correspond to (all functions vanishing on thecomplement of) regularly open sets, i.e., to open sets which equal the interior oftheir closure, and projection bands correspond to clopen sets. The Cantor set hasa basis of clopen sets, so that, in particular, every nonempty regularly open setcontains a nonempty clopen set. Therefore C(∆) has sufficiently many projections.It does not have the projection property, since [0, 1/4) ∩ ∆ ⊂ ∆ is regularly openbut not closed ([48, 29.7]).


Lemma 3.3.9. Let G be a finite group, E a Riesz space with sufficiently manyprojections, and ρ : G→ Aut+(E) a positive representation. Then the following areequivalent:

(i) ρ is order indecomposable;

(ii) ρ is projection band irreducible;

(iii) ρ is band irreducible.

Proof. (iii) ⇒ (ii) is immediate. For (ii) ⇒ (iii), suppose ρ is projection bandirreducible. Let B0 be a nonzero ρ-invariant band. Let 0 6= B ⊂ B0 be a projectionband. Then

∑s∈G ρsB is a projection band by [28, Theorem 30.1(ii)], and clearly

it is ρ-invariant, nonzero, and contained in B0. Therefore it must equal E, and soB0 = E.

(i)⇔ (ii) follows from Lemma 3.3.6.

This lemma will be improved significantly later on, see Theorem 3.3.16.We will now investigate the question whether a positive representation of a finite

group, satisfying a suitable notion of irreducibility, is necessarily finite dimensional.As explained in the introduction, this is not quite as obvious as it is for Banach spacerepresentations. It follows as a rather special case from [43, Theorem III.10.4] thata closed ideal irreducible positive representation of a finite group in a Banach latticeis finite dimensional, but this seems to be the only known available result in thisvein. We will show, see Theorem 3.3.14, that a positive principal band irreduciblerepresentation of a finite group in a Riesz space is finite dimensional. This impliesthe aforementioned finite dimensionality result for Banach lattices. As a preparation,we need four lemmas.

Lemma 3.3.10. Let G be a finite group, E a Riesz space and ρ : G → Aut+(E) apositive principal band irreducible representation. Then E is Archimedean.

Proof. Suppose E is not Archimedean. Then E 6= 0 and there exist x, y ∈ E suchthat 0 < λx ≤ y for all λ > 0. The band B generated by

∑s∈G ρsx is principal,

ρ-invariant and nonzero, and therefore equals E. Let u ≥ 0 be an element of theideal I generated by

∑s∈G ρsx. Then for some λ ≥ 0,

u ≤ λ∑s∈G

ρsx =∑s∈G

ρs(λx) ≤∑s∈G

ρsy,

and so∑s∈G ρsy is an upper bound for I+, and hence for B+ = E+, which is absurd

since E 6= 0. Therefore E is Archimedean.

Lemma 3.3.11. Let E be a Riesz space with dim(E) ≥ 2. Then E contains anontrivial principal band.

Proof. Suppose E does not contain a nontrivial principal band. Then the trivialgroup acts principal band irreducibly on E, so by Lemma 3.3.10, E is Archimedean.


Furthermore E is totally ordered, otherwise there exists an element x which is neitherpositive nor negative and so x+ /∈ Bx− = E. However, by [1, Exercise 1.14 (provenon page 272)], a totally ordered and Archimedean space has dimension 0 or 1, whichis a contradiction. Hence E contains a nontrivial principal band.

Lemma 3.3.12. Let E be an Archimedean Riesz space and let I ⊂ E be a finitedimensional ideal. Then I is a principal projection band.

Proof. By [28, Theorem 26.11] I ∼= Rn. Let e1, . . . , en be atoms that generate I. Itfollows that e1, . . . , en are also atoms in E, and that I =

∑k Iek , where Iek denotes

the ideal generated by ek. By [28, Theorem 26.4] the Iek are actually projectionbands in E, and so I, as a sum of principal projection bands, is a principal projectionband by [28, Chapter 4.31, page 181].

Lemma 3.3.13. Let G be a finite group, E an Archimedean Riesz space, B′ ⊂ E anonzero principal band and ρ : G→ Aut+(E) a positive representation. Then thereexists a nonzero principal band B ⊂ B′ such that for all t ∈ G, either B ∩ ρtB = 0or B = ρtB.

Proof. The set S := S ⊂ G : e ∈ S,⋂s∈S ρsB

′ 6= 0 is partially ordered byinclusion and nonempty, since e ∈ S. Pick a maximal element M ∈ S, and letB :=

⋂s∈M ρsB

′. Then B is a principal band by [28, Theorem 48.1]. Let t ∈ G andsuppose that B ∩ ρtB 6= 0. Then

0 6= B ∩ ρtB =⋂s∈M

ρsB′ ∩ ρt

⋂s∈M

ρsB′ =

⋂r∈M∪tM

ρrB′,

and by the maximality of M we obtain M ∪ tM = M , and so tM ⊂M . Combinedwith |tM | = |M | we conclude that tM = M , and then

ρtB = ρt⋂s∈M

ρsB′ =

⋂r∈tM

ρrB′ =

⋂r∈M

ρrB′ = B.

Using these lemmas, we can now establish our main theorem on finite dimen-sionality.

Theorem 3.3.14. Let G be a finite group, let E be a nonzero Riesz space and letρ : G → Aut+(E) a positive principal band irreducible representation. Then E isArchimedean, finite dimensional, and the dimension of E divides the order of G.

Proof. Lemma 3.3.10 shows that E is Archimedean. The proof is by induction on theorder of G. If G is the trivial group, then E is one dimensional by Lemma 3.3.11,and we are done. Suppose, then, that the theorem holds for all groups of orderstrictly smaller than the order or G. If E has only trivial principal bands, then byLemma 3.3.11 E has dimension one, and we are done again. Hence we may assumethat there exists a principal band 0 6= B′ 6= E. By Lemma 3.3.13 there exists a


nonzero principal band B ⊂ B′ 6= E such that H := t ∈ G : B = ρtB satisfiesHc = t ∈ G : B ∩ ρtB = 0. It is easy to see that H is a subgroup of G, and hasstrictly smaller order than G: otherwise B is a nontrivial ρ-invariant principal band,contradicting the principal band irreducibility of ρ.

We will now show that ρ restricted to H is principal band irreducible on theRiesz space B. Suppose 0 6= A ⊂ B is an H-invariant principal band of B. By [28,Theorem 48.1]

∑s∈G ρsA is a principal band, and so it is a nonzero ρ-invariant

principal band of E. Hence it equals E, so using [28, Theorem 20.2(ii)] in the secondstep and [53, Exercise 7.7(iii)] in the third step,

B = B ∩

∑s∈G

ρsA

=

B ∩

∑s∈G

ρsA

=

∑s∈G

(B ∩ ρsA)

=

∑s∈H

(B ∩ ρsA) +∑s∈Hc

(B ∩ ρsA)

⊂

∑s∈H

(B ∩ ρsA) +∑s∈Hc

(B ∩ ρsB)

⊂

∑s∈H

(B ∩A) +∑s∈Hc

0

=

∑s∈H

A

= A.

We conclude that ρ|H : H → Aut+(B) is principal band irreducible, so B has finitedimension by the induction hypothesis. By Lemma 3.3.12,

∑s∈G ρsB is a principal

band, which is nonzero and invariant, hence equal to E, and so E has finite dimensionas well.

Consider the sum∑sH∈G/H ρsB. This is well defined, since ρtB = B for

t ∈ H. Moreover, if sH 6= rH, then r−1s /∈ H and so ρr−1sB ∩ B = 0, imply-ing ρsB ∩ ρrB = 0. Therefore

∑sH∈G/H ρsB is a sum of ideals with pairwise zero

intersection, which is easily seen to be a direct sum using [28, Theorem 17.6(ii)]. Itfollows that

E =∑s∈G

ρsB =∑

sH∈G/H

ρsB =⊕

sH∈G/H

ρsB.


Therefore|G|

dim(E)=

|G||G : H|dim(B)

=|H|

dim(B)∈ N,

where the last step is by the induction hypothesis. Hence the dimension of E dividesthe order of G as well.

From Theorem 3.4.10, where we will explicitly describe all representations as inTheorem 3.3.14, it will also become clear that the dimension of the space dividesthe order of the group.

Remark 3.3.15. Note that Theorem 3.3.14 trivially implies a similar theorem forpositive representations which are ideal irreducible, or which are band irreducible.It also answers our original question as mentioned in the Introduction: a positiveprojection band irreducible representation of a finite group in a Banach lattice withthe projection property is finite dimensional. Indeed, an invariant principal band isthen an invariant projection band.

When combining Theorem 3.3.14 with Lemma 3.3.9, we obtain the followingresult. Amongst others it shows that, under a mild condition on the space, variousnotions of irreducibility for a positive representation of a finite group are, in fact,the same for finite groups. It should be compared with the equality of irreducibilityand indecomposability for unitary representations of arbitrary groups, and for finitedimensional representations of finite groups whenever Maschke’s theorem applies.As already mentioned earlier, if a Riesz space has sufficiently many projections, itis automatically Archimedean, cf. [28, Theorem 30.4].

Theorem 3.3.16. Let G be a finite group, E a Riesz space with sufficiently manyprojections and ρ : G → Aut+(E) a positive representation. Then the following areequivalent:

(i) ρ is order indecomposable;

(ii) ρ is projection band irreducible;

(iii) ρ is band irreducible;

(iv) ρ is ideal irreducible;

(v) ρ is principal band irreducible.

If these equivalent conditions hold, then E is finite dimensional, and the dimensionof E divides the order of G if E is nonzero.

Proof. By Lemma 3.3.9 the first three conditions are equivalent. Each of the lastthree conditions implies that ρ is principal band irreducible, so by Theorem 3.3.14each of these three conditions implies that E is finite dimensional, hence latticeisomorphic to Rn for some n ([28, Theorem 26.11]). But then the collections of bands,ideals and principal bands in E are all the same, and hence the last three conditionsare equivalent as well. The remaining statements follow from Theorem 3.3.14.

3.4. STRUCTURE OF POSITIVE REPRESENTATIONS 89

3.4 Structure of finite dimensional positiveArchimedean representations

Now that we know from Section 3.3 that positive representations of finite groups,irreducible as in Theorem 3.3.14 or 3.3.16, are necessarily in Archimedean and finitedimensional spaces, our goal is to describe the general positive finite dimensionalArchimedean representations of a finite group. In such spaces, the collections of(principal) ideals, (principal) bands and projection bands are all the same, andwe will use the term “irreducible positive representation” throughout this section todenote the corresponding common notion of irreducibility, which is the same as orderindecomposability. We will see in Theorem 3.4.9 that positive finite dimensionalArchimedean representations of a finite group split uniquely into irreducible positiverepresentations. Furthermore, the order equivalence classes of finite dimensionalirreducible positive representations are in natural bijection with the isomorphismclasses of transitive G-spaces, cf. Theorem 3.4.10. The latter result can be thoughtof as the description of the finite dimensional Archimedean part of the order dualof a finite group. The fact that such irreducible positive representations can berealized in this way also follows from [43, Theorem III.10.4], where it is shownthat strongly continuous closed ideal irreducible positive representations of a locallycompact group in a Banach lattice, with compact image in the strong operatortopology, can be realized on function lattices on homogeneous spaces. This generalresult, however, requires considerable machinery. Therefore we prefer the methodbelow, where all follows rather easily once an explicit description of the general, notnecessarily irreducible, positive representation of a finite group in a finite dimensionalArchimedean space has been obtained, a result which has some relevance of its own.

Since the decomposition result below is such a close parallel to classical semisim-ple representation theory of finite groups, it is natural to ask whether any otherfeatures of this purely linear context survive, such as character theory. At the endof this section we show that this is, for general groups, not the case, and in the nextsection we will see that this is only partly so for induction.

We now proceed towards the first main step, the description of a positive finite di-mensional Archimedean representation of a finite group. Since an Archimedean finitedimensional Riesz spaces is isomorphic to Rn for some n ([28, Theorem 26.11]), westart by describing its group Aut+(Rn) of lattice automorphisms. Naturally, the wellknown result [32, Theorem 3.2.10] on lattice homomorphisms between C0(K)-spacesdirectly implies the structure of Aut+(Rn), but in this case, where K = 1, . . . , n,this can be seen in an elementary fashion as below. Subsequently we determine thefinite subgroups of Aut+(Rn). After that, we can describe the positive representa-tions of a finite group in Rn and continue from there.

3.4.1 Description of Aut+(Rn)

We denote the standard basis of Rn by e1, . . . , en. A lattice automorphism mustobviously map positive atoms to positive atoms, so each T ∈ Aut+(Rn) maps ei to


λjiej for some λji > 0. This implies that T can be written uniquely as the productof a strictly positive multiplication (diagonal) operator and a permutation operator.We identify the group of permutation operators with Sn, so each σ ∈ Sn correspondsto the operator mapping ei to eσ(i). The group of strictly positive multiplicationoperators is identified with (R>0)n, and so there exist unique m ∈ (R>0)n andσ ∈ Sn such that T = mσ.

For σ ∈ Sn and m ∈ (R>0)n, define σ(m) ∈ (R>0)n by σ(m)i := mσ−1(i). Thisdefines a homomorphism of Sn into the automorphism group of (R>0)n, hence wecan form the corresponding semidirect product (R>0)n o Sn, with group operation(m1, σ1)(m2, σ2) := (m1σ1(m2), σ1σ2). On noting that, for i = 1, . . . , n,

σmσ−1ei = σmeσ−1(i) = σmσ−1(i)eσ−1(i) = mσ−1(i)ei = σ(m)iei = σ(m)ei,

it follows easily that χ : (R>0)n o Sn → Aut+(Rn), defined by χ(m,σ) := mσ, isa group isomorphism. From now on we identify Aut+(Rn) and (R>0)n o Sn, usingeither the operator notation or the semidirect product notation.

We let p : Aut+(Rn)→ Sn, defined by p(m,σ) := σ, denote the canonical homo-morphism of the semidirect product onto the second factor.

3.4.2 Description of the finite subgroups of Aut+(Rn)

Let G be a finite subgroup of Aut+(Rn). Then ker(p|G) can be identified with afinite subgroup of ker(p) = (R>0)n. Clearly the only finite subgroup of (R>0)n

is trivial, and so p|G is an isomorphism. It follows that every finite subgroup ofAut+(Rn) is isomorphic to a finite subgroup of Sn. The next proposition makes thiscorrespondence explicit.

Proposition 3.4.1. Let A be the set of finite subgroups G ⊂ Aut+(Rn), and let Bbe the set of pairs (H, q), where H ⊂ Sn is a finite subgroup and q : H → Aut+(Rn)is a group homomorphism such that p q = idH . Define α : A → B and β : B → Aby

α(G) :=(p(G), (p|G)−1

), β(H, q) := q(H).

Then α and β are inverses of each other.

Proof. Clearly p (p|G)−1 = idp(G), so α is well defined. Let G ∈ A, then

β(α(G)) = β(p(G), (p|G)−1) = G.

Conversely, let (H, q) ∈ B, then α(β(H, q)) = α(q(H)) = (p(q(H)), (p|q(H))−1), and

since p q = idH , it follows that p(q(H)) = H and that

(p|q(H))−1 = (p|q(H))

−1 p q = q.


By the above proposition each finite subgroup G of Aut+(Rn) is determined bya subgroup H of Sn and a homomorphism q : H → Aut+(Rn) satisfying p q = idH .We will now investigate such maps q. The condition p q = idH is equivalent withthe existence of a map f : H → (R>0)n, such that q(σ) = (f(σ), σ) for σ ∈ H. Forσ, τ ∈ H, we have q(στ) = (f(στ), στ) and

q(σ)q(τ) = (f(σ), σ)(f(τ), τ) = (σ(f(τ))f(σ), στ).

Hence q being a group homomorphism is equivalent with f(στ) = σ(f(τ))f(σ) forall σ, τ ∈ H, and such maps are called crossed homomorphisms.

Crossed homomorphisms of a finite group into a suitably nice abelian group(A,+) (in our case ((R>0)n, ·)) can be characterized by the following lemma, whichstates, in the language of group cohomology, that H1(H,A) is trivial.

Lemma 3.4.2. Let H be a finite group acting on an abelian group (A,+) suchthat, for all a ∈ A, there exists a unique element of H, denoted by a/|H|, satisfying|H|(a/|H|) = a. Let f : H → A be a map. Then f is a crossed homomorphism, i.e.,f(st) = s(f(t)) + f(s) for all s, t ∈ H, if and only if there exists an a ∈ A such that

f(s) = a− s(a) ∀s ∈ H.

Proof. Suppose f is a crossed homomorphism. Let a := 1|H|∑r∈H f(r), then, for

s ∈ H,

s(a) =1

|H|∑r∈H

s(f(r))

=1

|H|∑r∈H

[f(sr)− f(s)]

=1

|H|∑r∈H

[f(r)− f(s)]

= a− f(s).

Hence f(s) = a− s(a), as required. The converse is trivial.

Combining this result with the previous discussion, we obtain the following.

Corollary 3.4.3. Let H be a finite subgroup of Sn and let q : H → Aut+(Rn) be amap. Then q is a homomorphism satisfying p q = idH if and only if there existsan m ∈ (R>0)n such that

q(σ) = (mσ(m)−1, σ) ∀σ ∈ H.

Rewriting this in multiplicative notation rather than semidirect product notationyields the following.


Corollary 3.4.4. Let G be a finite subgroup of Aut+(Rn). Then there is a uniquefinite subgroup H ⊂ Sn and an m ∈ (R>0)n such that

G =mσ(m)−1σ : σ ∈ H

= mHm−1.

Conversely, if H ⊂ Sn is a finite subgroup and m ∈ (R>0)n, then G ⊂ Aut+(Rn)defined by the above equation is a finite subgroup of Aut+(Rn).

Proof. By Proposition 3.4.1, G = q(p(G)), for some q : p(G) → (R>0)n satisfyingp q = idp(G). So H = p(G) is unique, and the rest follows from the previouscorollary.

Note that, given a finite subgroup G ⊂ Aut+(Rn), the subgroup H ⊂ Sn isunique, but the multiplication operator m in Corollary 3.4.4 is obviously not unique,e.g., both m and λm for λ > 0 induce the same G.

3.4.3 Positive finite dimensional Archimedeanrepresentations

In this subsection we obtain our main results on finite dimensional positive rep-resentations of finite groups in Archimedean spaces: explicit description of suchrepresentations (Theorem 3.4.5), decomposition into irreducible positive representa-tions (Theorem 3.4.9) and description of irreducible positive representations up toorder equivalence (Theorem 3.4.10).

Applying the results from the previous subsection, in particular Proposition 3.4.1and Lemma 3.4.2, we obtain the following. Recall that we view Sn ⊂ Aut+(Rn), byidentifying σ ∈ Sn with a permutation matrix, and a representation π : G → Sn iscalled a permutation representation.

Theorem 3.4.5. Let G be a finite group and ρ : G → Aut+(Rn) a positive repre-sentation. Then there is a unique permutation representation π and an m ∈ (R>0)n

such thatρs = mπsm

−1 ∀s ∈ G.Conversely, any permutation representation π : G → Sn and m ∈ (R>0)n define apositive representation ρ by the above equation.

Proof. Applying Proposition 3.4.1 to ρ(G) and combining this with Lemma 3.4.2,p : ρ(G)→ pρ(G) has an inverse of the form q(σ) = mσ(m)−1σ for somem ∈ (R>0)n

and all σ ∈ p ρ(G). We define π := p ρ, then for s ∈ G,

ρs = (q p)(ρs) = q(πs) = mπs(m)−1πs = mπsm−1.

This shows the existence of π. The uniqueness of π follows from the uniqueness ofthe factors in (R>0)n and Sn in

ρs = mπsm−1 = [mπs(m)−1]πs.

The converse is clear.


If ρ, π and m are related as in the above theorem, we will denote this asρ ∼ (m,π). Note that, as in Corollary 3.4.4, π is unique, but m is not. Giventhe permutation representation π, m1 and m2 induce the same positive representa-tions if and only if m1m

−12 = πs(m1m

−12 ) for all s ∈ G.

Recall that if ρ, θ : G → Aut+(Rn) are positive representations, we call themorder equivalent if there exists an intertwiner T ∈ Aut+(R>0)n between ρ and θ.We call them permutation equivalent if there exists an intertwiner σ ∈ Sn; thisimplies order equivalence.

Proposition 3.4.6. Let G be a finite group and ρ1 ∼ (m1, π1) and ρ2 ∼ (m2, π

2)be two positive representations of G in Rn. Then ρ1 and ρ2 are order equivalent ifand only if π1 and π2 are permutation equivalent.

Proof. Suppose that ρ1 and ρ2 are order equivalent and let T = (m,σ) ∈ Aut+(Rn)be an intertwiner. Then, for all s ∈ G,

ρ1sT = (m1π

1s(m1)−1, π1

s)(m,σ) = (m1π1s(m1)−1π1

s(m), π1sσ) (3.4.1)

Tρ2s = (m,σ)(m2π

2s(m2)−1, π2

s) = (mσ(m2)σπ2s(m2)−1, σπ2

s), (3.4.2)

and since these are equal, σ is an intertwiner between π1 and π2.Conversely, let σ be an intertwiner between π1 and π2. Then, by choosing

m = m1σ(m2)−1 and T = (m,σ) ∈ Aut+(Rn), it is easily verified that, for alls ∈ G,

(m1π1s(m1)−1π1

s(m), π1sσ) = (mσ(m2)σπ2

s(m2)−1, σπ2s),

and so by (3.4.1) and (3.4.2), T intertwines ρ1 and ρ2.

We immediately obtain that every positive representation is order equivalent toa permutation representation.

Corollary 3.4.7. Let G be a finite group and let ρ ∼ (m,π) be a positive represen-tation of G in Rn. Then ρ is order equivalent to (1, π).

Remark 3.4.8. The method in this subsection also yields a description of thehomomorphisms from a finite group into a semidirect product NoK with N torsion-free and H1(H,N) trivial for all finite subgroups H ⊂ K, but we are not aware of areference for this fact.

Using Corollary 3.4.7, we obtain our decomposition theorem.

Theorem 3.4.9. Let G be a finite group and ρ : G→ Aut+(E) a positive represen-tation in a nonzero finite dimensional Archimedean Riesz space E. Let Bii∈I bethe set of irreducible invariant bands in E. Then E = ⊕i∈IBi. Furthermore, anyinvariant band is a direct sum of Bi’s.

Proof. We may assume that E = Rn, where bands are just linear spans of a numberof standard basis vectors. By Corollary 3.4.7, we may assume that ρ is a permutationrepresentation, which is induced by a group action on the set of basis elements. It


is then clear that the irreducible invariant bands correspond to the orbits of thisgroup action, and the invariant bands to unions of orbits. This immediately givesthe decomposition of ρ into irreducible positive representations, and the descriptionof the invariant bands.

We will now give a description of what could be called the finite dimensionalArchimedean part of the order dual of a finite group. Note that Theorems 3.3.14and Theorem 3.3.16 imply that a number of positive representations, irreduciblein an appropriate way, are automatically in finite dimensional Archimedean spaces.Hence they fall within the scope of the next result, which is formulated in terms ofa function lattice in order to emphasize the similarity with [43, Theorem III.10.4].

Theorem 3.4.10. Let G be a finite group. If H ⊂ G is a subgroup, let (etH)tH∈G/Hbe the canonical basis for the finite dimensional Riesz space C(G/H), defined byetH(sH) = δtH,sH , for tH, sH ∈ G/H. Let πH : G → Aut+(C(G/H)) be thecanonical positive representation corresponding to the action of G on G/H, so thatπHs etH := estH for s, t ∈ G. Then, whenever H1 and H2 are conjugate, πH1 andπH2 are order equivalent, and the map

[H] 7→ [πH ]

is a bijection between the conjugacy classes of subgroups of G and the order equiva-lence classes of irreducible positive representations of G in nonzero finite dimensionalArchimedean Riesz spaces.

Proof. It follows easily from Theorem 3.2.2 that the map is well-defined. As a con-sequence of Corollary 3.4.7, every nonzero finite dimensional positive Archimedeanrepresentation is order equivalent to a permutation representation, arising from anaction of G on 1, . . . , n for some ≥ 1. Since the irreducibility of π is then equiva-lent to the transitivity of this group action, this shows that the map is surjective.As to injectivity, suppose that πH1 and πH2 are order equivalent, for subgroupsH1, H2 of G. Let n = |G : H1| = |G : H2|, and consider Rn with standard ba-sis e1, . . . , en. Choose a bijection between the canonical basis for C(G/H1) ande1, . . . , en, and likewise for the canonical basis of C(G/H2). This gives a latticeisomorphism between C(G/H1) and Rn, and similarly for C(G/H2). After transportof structure, G has two positive representations on Rn which are order equivalentby assumption, and which originate from two permutation representations on thesame set 1, . . . , n. As a consequence of Proposition 3.4.6, the permutation partsof these positive representations are permutation equivalent, i.e., the two G-spaces,consisting of 1, . . . , n and the respective G-actions, are isomorphic G-spaces. Con-sequently, G/H1 and G/H2 are isomorphic G-spaces, and then Lemma 3.2.2 showsthat H1 and H2 are conjugate.

If n ∈ Z≥0 and ρ is a positive representation, then nρ denotes the n-fold orderdirect sum of ρ. Combining the above theorem with Theorem 3.4.9, we immediatelyobtain the following, showing how the representations under consideration are builtfrom canonical actions on function lattices on transitive G-spaces.


Corollary 3.4.11. Let G be a finite group and let H1, . . . ,Hk be representatives ofthe conjugacy classes of subgroups of G. Let E be a finite dimensional ArchimedeanRiesz space and let ρ : G → Aut+(E) be a positive representation. Then, using thenotation of Theorem 3.4.10, there exist unique n1, . . . , nk ∈ Z≥0 such that ρ is orderequivalent to

n1πH1 ⊕ · · · ⊕ nkπHk .

3.4.4 Linear equivalence and order equivalence

If two unitary group representations are intertwined by a bounded invertible oper-ator, they are also intertwined by a unitary operator [10, Section 2.2.2]. We willnow investigate the corresponding natural question in the finite dimensional orderedsetting: if two positive finite dimensional Archimedean representations are inter-twined by an invertible linear map, are they order equivalent? By character theory,see for example [25, Theorem XVIII.2.3], two representations over the real numbersare linearly equivalent if and only if they have the same character. The followingexample, taken from the introduction of [26], therefore settles the matter.

Example 3.4.12. Let G be the group Z/2Z × Z/2Z, and conser the permutationrepresentations π1, π2 : G→ Aut+(R6) determined by

π1(0,1) := (12)(34) π1

(1,0) := (13)(24) π2(0,1) := (12)(34) π2

(1,0) := (12)(56).

Then, as is easily verified, π1 and π2 have the same character, and so they arelinearly equivalent. However, they are not order equivalent, since by examining theorbits of standard basis elements it follows that the first representation splits intothree irreducible positive representations of dimensions 1, 1, and 4, and the secondsplits into three irreducible positive representations of dimension 2 each.

Thus, in general, linear equivalence (equivalently: equality of characters) of posi-tive representations does not imply order equivalence. One might then try to narrowdown the field: is perhaps true that two irreducible positive representations, whichare linearly equivalent, are order equivalent? In view of Theorem 3.4.10 and Theo-rem 3.2.2, this is asking whether a linear equivalence of the positive representationscorresponding to two transitive G-spaces (which is equivalent to equality, for eachgroup element, of the number of fixed points in the two spaces) implies that these G-spaces are isomorphic. The answer, again, is negative, but counterexamples are nowmore intricate to construct than above, and the reader is referred to [47, Theorem 1],providing an abundance of such counterexamples.

On the positive side, in some cases linear equivalence of irreducible positiverepresentations does imply order equivalence, as shown by the next result.

Lemma 3.4.13. Let G be a finite group, let N be a normal subgroup, and letπN : G → Aut+(C(G/N)) be the irreducible positive representations as in Theo-rem 3.4.10. Then an irreducible positive representation which is linearly equivalentwith πN is in fact order equivalent with πN .


Proof. Passing to an order equivalent model we may, in view of Theorem 3.4.10,assume that the other irreducible positive representation is πH , for a subgroup Hof G. The fact that G/N is a group implies easily that the character of πN equals|G : N |1N . Since the character of πH , which is equal to that of πN by their linearequivalence, is certainly nonzero on H, we see that H ⊂ N . On the other hand,equality of dimensions yields |G : N | = |G : H|, hence |N | = |H|. We conclude thatH = N .

Combining Theorem 3.4.10 with the previous lemma yields the following.

Corollary 3.4.14. Let G be a finite group with only normal subgroups. If twofinite dimensional irreducible positive representations of G are linearly equivalent(equivalently: have the same character), they are order equivalent.

Thus, for such groups (so-called Dedekind groups), and in particular for abeliangroups, the classical correspondence between characters and irreducible representa-tions survives in an ordered context—where, naturally, “irreducible” has a differentmeaning. However, as Example 3.4.12 shows, already for abelian groups this corre-spondence breaks down for reducible positive representations.

3.5 Induction

In this section we will examine the theory of induction in the ordered setting froma categorical point of view. It turns out that the results are to a large extentanalogous to the linear case as covered in many sources, e.g., [25, Section XVIII.7].Still, it seems worthwhile to go through the motions, as a preparation for futuremore analytical constructions, and in doing so we then also obtain a slight bonus (theoriginal ordered module is embedded in the induced one as a sublattice, even thoughthis was not required), keep track of several notions of irreducibility, and also observethat Frobenius reciprocity holds only partially. Since we do not consider topologicalissues at the moment, we are mostly interested in the case where the group is finite,but the theory is developed at little extra cost in general for arbitrary groups andsubgroups. Our approach is thus slightly more general than, e.g., the approach in[25], as we do not require our groups to be finite or our subgroups to be of finiteindex.

For the rest of the section, G is a not necessarily finite group, H is a subgroup ofG, not necessarily finite or of finite index, R is a system of representatives of G/H,and the Riesz spaces are not assumed to be finite dimensional. The only finitenesscondition is in Corollary 3.5.11, where G is assumed to be finite.

3.5.1 Definitions and basic properties

A pair (E, ρ), where E is a Riesz space and ρ : G → Aut+(E) is a positive repre-sentation, is called an ordered G-module. In this notation, we will often omit therepresentation ρ. If E is an ordered G-module, it is also an ordered H-module by

3.5. INDUCTION 97

restricting the representation to H. If (E, ρ) and (F, θ) are ordered G-modules,then the positive cone of positive intertwiners between ρ and θ will be denoted byHom+

G(E,F ). Two ordered G-modules are isomorphic ordered G-modules if thereexists an intertwining lattice isomorphism.

Definition 3.5.1. Let F be an ordered H-module. A pair (IndGH(F ), j), whereIndGH(F ) is an ordered G-module and j ∈ Hom+

H(F, IndGH(F )) is called an inducedordered module of F from H to G if it satisfies the following universal property:

For any ordered G-module E and T ∈ Hom+H(F,E), there is a unique map

T ∈ Hom+G(IndGH(F ), E) such that T = T j, i.e., such that the following diagram

is commutative:

F

j ##GGGGGGGGGT // E

IndGH(F )

T

;;wwwwwwwww

If θ is the positive representation of H turning F into an ordered H-module, thenthe positive representation of G turning IndGH(F ) into an ordered G-module will bedenoted by IndGH(θ) and will be called the induced positive representation of θ fromH to G.

First we will show, by the usual argument, that the induced ordered module isunique up to isomorphism of ordered G-modules.

Lemma 3.5.2. Let F be an ordered H-module and let (E1, j1) and (E2, j2) be in-duced ordered modules of F from H to G. Then E1 and E2 are isomorphic as orderedG-modules.

Proof. Using the universal property, we obtain the unique maps j1 ∈ Hom+G(E2, E1)

satisfying j1 = j1 j2 and j2 ∈ Hom+G(E1, E2) satisfying j2 = j2 j1. It follows that

j1 = j1 j2 = j1 j2 j1.

Now consider the orderedG-module E1, and apply its universal property to itself - weobtain the unique map idE1

∈ Hom+G(E1, E1) such that j1 = idE1

j1. Together withthe above equation, this shows that j1 j2 = idE1

. Similarly we obtain j2 j1 = idE2,

and so j2 is an isomorphism of ordered G-modules.

We will now construct the induced ordered module, which is the usual inducedmodule, but now with an additional lattice structure. Let (F, θ) be an orderedH-module. We define the ordered vector space

E := f : G→ F | f(st) = θt−1f(s) ∀s ∈ G, ∀t ∈ H, (3.5.1)

with pointwise ordering. Using that θt−1 is a lattice isomorphism for t ∈ H, oneeasily verifies that E is a Riesz space, with pointwise lattice operations. Furthermore,if S ⊂ G is a subset, then we define the subset

ES := f ∈ E | supp (f) ⊂ S. (3.5.2)


Let ρ : G → Aut+(E) be defined by (ρsf)(u) := f(s−1u), for s, u ∈ G. Then ρ is apositive representation turning E into an ordered G-module. Moreover, for s ∈ G,supp (ρsf) = s · supp (f), so ρsEH = EsH . Now we define the ρ-invariant Rieszsubspace

E :=⊕r∈R

ErH =⊕r∈R

ρrEH ⊂ E. (3.5.3)

For x ∈ F , define a function j(x) : G → F by j(x)(t) := θt−1x for t ∈ H, andj(x)(t) := 0 for t /∈ H. It is routine to verify that j(x) ∈ EH for all x ∈ F , and thatj : F → EH is an isomorphism of ordered H-modules between (F, θ) and (EH , ρ).This last fact and the fact that E = ⊕r∈RρrF as an order direct sum will be usedto prove that (E, j) actually satisfies the desired universal property.

Lemma 3.5.3. Let (E, ρ) be an ordered G-module and let F be a Riesz subspaceof E which is invariant under the restricted representation θ = ρ|H of H in E, andsuch that E = ⊕r∈RρrF as an order direct sum. Let j : F → E be the embedding.Then (E, j) is an induced ordered module of F from H to G.

Proof. Let (E′, ρ′) be another ordered G-module, and let T : F → E′ be a positivelinear map such that T (θtx) = ρ′tT (x) for all t ∈ H and x ∈ F . We have to showthat there exists a unique positive linear map T : E → E′ extending T and satisfyingT ρs = ρ′s T for all s ∈ G.

We follow the proof of [45, Lemma 3.3.1]. If T satisfies these conditions, and ifx ∈ ρrF for r ∈ R, then ρ−1

r x ∈ F , and so

T (x) = T (ρrρ−1r x) = ρ′rT (ρ−1

r x) = ρ′rT (ρ−1r x).

This formula determines T on ρrF , hence on E since it is the direct sum of the ρrF .This proves the uniqueness of T .

For the existence, let x ∈ ρrF , then we define T (x) := ρ′rT (ρ−1r x) as above. This

does not depend on the choice of representative r, since if we replace r by rt witht ∈ H, we have

ρ′rtT (ρ−1rt x) = ρ′rρ

′tT (θ−1

t ρ−1r x) = ρ′rT (θtθ

−1t ρ−1

r x) = ρ′rT (ρ−1r x).

Since E is the direct sum of the ρrF , there exists a unique linear map T : E → E′

which extends the partial mappings thus defined on the ρrF . It is easily verifiedthat Tρs = ρ′sT for all s ∈ G. Since all mappings involved are positive, T is positiveas well.

Corollary 3.5.4. Let F be an ordered H-module. Then the induced ordered module(IndGH(F ), j) of F from H to G exists, and IndGH(F ) is unique up to isomorphism ofordered G-modules. The positive map j is actually an injective lattice homomorphismof the ordered H-module F into the ordered H-module IndGH(F ). Moreover, if E isfinite dimensional and H has finite index, then dim(IndGH(F )) = |G : H|dim(E).

Proof. The existence follows from Lemma 3.5.3 and the construction preceding it,which also shows that j has the property as described. The uniqueness follows fromLemma 3.5.2. The last statement follows from (3.5.3).

3.5. INDUCTION 99

We continue with some properties of the induced positive representation.

Proposition 3.5.5. Let θ be a positive representation of a subgroup H ⊂ G. IfIndGH(θ) is either band irreducible, or ideal irreducible, or projection band irreducible,then so is θ.

Proof. We will prove this for ideals, the other cases are identical. Let E be as in(3.5.1), (3.5.2) and (3.5.3). Suppose θ is not ideal irreducible, so there exists a propernontrivial θ-invariant ideal B ⊂ Eθ. Then f ∈ E : f(G) ⊂ B ⊂ E is a propernontrivial IndGH(θ)-invariant ideal, so IndGH(θ) is not ideal irreducible.

Proposition 3.5.6 (Induction in stages). Let H ⊂ K ⊂ G be a chain of subgroupsof G, and let F be an ordered H-module. Then

(IndGK(IndKH(F )), jGK jKH )

is an induced ordered module of F from H to G.

Proof. Let E be an ordered G-module. Consider the following diagram:

FjKH //

T

##GGGGGGGGGG IndKH(F )jGK //

T

IndGK(IndKH(F ))

Twwnnnnnnnnnnnnn

E

Here T is the unique positive map generated by T , and T is the unique positive

map generated by T . Since the diagram is commutative, T = T (jGK jKH ). If S isanother positive map satisfying T = S (jGK jKH ), then (S jGK) jKH = T = T jKH ,

and so S jGK = T by the uniqueness of T . This in turn implies that S = T by the

uniqueness of T , and so (IndGK(IndKH(F )), jGK jKH ) satisfies the universal property,as desired.

3.5.2 Frobenius reciprocity

This subsection is concerned with the implications, or rather their absence, of thefunctorial formulation of Frobenius reciprocity for multiplicities of irreducible posi-tive representations in induced ordered modules.

We start with the usual consequence of the categorical definition of the inducedordered module: induction from H to G is the left adjoint functor of restriction fromG to H, for an arbitrary group G and subgroup H.

Proposition 3.5.7 (Frobenius Reciprocity). Let F be an ordered H-module and letE be an ordered G-module. Then there is a natural isomorphism of positive cones

Hom+H(F,E) ∼= Hom+

G(IndGH(F ), E).


Proof. The existence of the natural bijection is an immediate consequence of thevery definition of the induced module in Definition 3.5.1. For T, S ∈ Hom+

H(F,E)and λ, µ ≥ 0 we have λT + µS = λT + µS as a consequence of the uniqueness partof Definition 3.5.1 and the positivity of λT + µS. Hence the two positive cones areisomorphic.

Now supposeG is a finite group. For finite dimensional positive Archimedean rep-resentations of G we have a unique decomposition into irreducible positive represen-tations, according to Corollary 3.4.11. Hence the notion of multiplicity is available,and if ρ1 is a finite dimensional positive representation and ρ2 a finite dimensionalirreducible positive representation of G, we let m(ρ1, ρ2) denote the number of timesthat a lattice isomorphic copy of ρ1 occurs in the decomposition of ρ2 of Corol-lary 3.4.11. Now Let θ be a finite dimensional irreducible positive representation ofa subgroup H ⊂ G and let ρ be a finite dimensional irreducible positive represen-tation of G. In view of the purely linear theory, cf. [45, Proposition 21], and itsgeneralization to unitary representations of compact groups, cf. [52, Theorem 5.9.2],it is natural to ask whether

m(ρ, IndGH(θ)) = m(θ, ρ|H)

still holds in our ordered context. In the linear theory, and also for compact groups,this follows from the fact that the dimensions of spaces of intertwining operatorsin the analogues of Proposition 3.5.7 can be interpreted as a multiplicities. Sincethe sets in Proposition 3.5.7 are cones and not vector spaces, and their elementsare not even necessarily lattice homomorphisms, there seems little chance of suc-cess in our case. Indeed, Frobenius reciprocity in terms of multiplicities does nothold for ordered modules, as is shown by the following counterexample. We letθ : H = e → Aut+(R) be the trivial representation; then IndGH(θ) is the left reg-ular representation of G on the Riesz space with atomic basis ess∈G. This set ofbasis elements has only one G-orbit, hence IndGH(θ) is band irreducible. Therefore,if ρ is an arbitrary irreducible positive representation of G, m(ρ, IndGH(θ)) is at mostone. On the other hand, ρ|H decomposes as dim ρ copies of the trivial representationθ, so m(θ, ρ|H) = dim ρ.

Another counterexample, where H is nontrivial, can be obtained by takingG = Z/4Z and H = 0, 2, and taking θ and ρ to be the left regular representationof H and G, respectively. Then these are irreducible positive representations of therespective groups, and it can be verified that IndGH(θ) ∼= ρ, so m(ρ, IndGH(θ)) = 1,but ρ|H ∼= θ ⊕ θ, so m(θ, ρ|H) = 2.

3.5.3 Systems of imprimitivity

In this final subsection, we consider systems of imprimitivity in the ordered setting.As before, G is an arbitrary group and H ⊂ G an arbitrary subgroup. We start withan elementary lemma, which is easily verified.

3.5. INDUCTION 101

Lemma 3.5.8. Let E and F be Riesz spaces and let T : E → F be a lattice isomor-phism. If B ⊂ E is a projection band in E, then TB is a projection band in F , andthe corresponding band projections are related by PTB = TPBT

−1.

Let ρ : G → Aut+(E) be a positive representation. Suppose there exists a G-space Γ, and a family of Riesz subspaces Eγγ∈Γ, such that E = ⊕γ∈ΓEγ as anorder direct sum and ρsEγ = Esγ for all s ∈ G. Then we call the family Eγγ∈Γ

an ordered system of imprimitivity for ρ. If A ⊂ Γ, then the order decomposition

E =

⊕γ∈A

Eγ

⊕⊕γ∈Ac

Eγ

implies that ⊕γ∈AEγ is a projection band. In particular, Eγ is a projection bandfor all γ ∈ Γ. For a subset A ⊂ Γ, let P (A) denote the band projection P⊕γ∈AEγonto ⊕γ∈AEγ . Then the assignment A 7→ P (A) is a band projection valued mapsatisfying

P

(⋃i∈I

Ai

)x = sup

i∈IP (Ai)x (3.5.4)

for all x ∈ E+ and all collections of subsets Aii∈I ⊂ Γ. An equivalent formulationof (3.5.4) is P (supiAi) = supi P (Ai), where the first supremum is taken in thepartially ordered set of subsets of Γ, and the second supremum is taken in thepartially ordered space of regular operators on E. Either formulation is the orderedanalogue of a strongly σ-additive spectral measure. Furthermore, the map P iscovariant in the sense that, for s ∈ G,

P (sA) = P⊕γ∈AEsγ = Pρs⊕γ∈AEγ = ρs(P⊕γ∈AEγ

)ρ−1s = ρsP (A)ρ−1

s ,

where the above lemma is used in the penultimate step. Every positive representa-tion admits a system of imprimitivity where Γ has exactly one element, and sucha system of imprimitivity will be called trivial. A system of imprimitivity is calledtransitive if the action of G on Γ is transitive.

Definition 3.5.9. A positive representation ρ is called primitive if it admits onlythe trivial ordered system of imprimitivity.

Every decomposition of E into ρ-invariant projection bands corresponds to an or-dered system of imprimitivity where the action of G on Γ is trivial, so ρ is projectionband irreducible if and only if ρ admits no nontrivial ordered system of imprimitivitywith a trivial action. Hence a primitive positive representation is projection bandirreducible, i.e., order indecomposable.

Theorem 3.5.10 (Imprimitivity Theorem). Let ρ be a positive representation of G.The following are equivalent:

(i) ρ admits a nontrivial ordered transitive system of imprimitivity;


(ii) There exists a proper subgroup H ⊂ G and a positive representation θ of Hsuch that ρ is order equivalent to IndGH(θ).

Proof. (ii) ⇒ (i): Suppose that ρ is order equivalent to IndGH(θ). Let Γ be thetransitive G-space G/H, which is nontrivial because H is proper, and consider thespaces E and EsHsH∈Γ defined in (3.5.1), (3.5.2) and (3.5.3). By the discussionfollowing these definitions, E = ⊕sH∈ΓEsH , and ρuEsH = EusH , so this defines anontrivial transitive system of imprimitivity.

(i) ⇒ (ii): Suppose ρ admits a nontrivial transitive ordered system of imprim-itivity. Then by Theorem 3.2.2 we may assume Γ = G/H for some subgroup H,which must be proper since Γ is nontrivial. Choose a system of representatives Rof G/H, then we may assume Γ = R. Assume that the representative of H in Ris e. We have that E = ⊕r∈REr, and if t ∈ H, then t acts trivially on e ∈ R, soρtEe = Ee. Therefore we can define θ : H → Aut+(Ee) by restricting ρ to H andletting it act on Ee. Then by the definition of the system of imprimitivity Err∈Rwe have ρrEe = Er, and so ⊕

r∈RρrEe =

⊕r∈R

Er = E,

which implies by Lemma 3.5.3 that ρ is induced by θ.

Corollary 3.5.11. All projection band irreducible positive representations of a finitegroup G are induced by primitive positive representations.

Proof. Let ρ be a projection band irreducible representation. If ρ is primitive we aredone, so assume it is not primitive. Then there exists a nontrivial ordered systemof imprimitivity Eγγ∈Γ. Then for each orbit in Γ, the direct sum of Eγ , whereγ runs through the orbit, is a ρ-invariant projection band. Since ρ is projectionband irreducible, this implies that Γ is transitive. Therefore, by the ImprimitivityTheorem 3.5.10, ρ is induced by a positive representation of a proper subgroup of G,which is projection band irreducible by Proposition 3.5.5. If this representation isprimitive we are done, and if not we keep repeating the process until a representationis induced by a primitive positive representation. Then by Proposition 3.5.6 therepresentation ρ is induced by this primitive positive representation as well.

We note that, in the above corollary, the representations need not be finite di-mensional, and that it trivially implies a similar statement for band irreducible andideal irreducible positive representations of a finite group.

Acknowledgements

The authors thank Lenny Taelman for helpful comments and suggestions.

Chapter 4

Compact groups of positiveoperators on Banach lattices

This chapter has been submitted for publication as: M. de Jeu and M. Wortel,“Compact groups of positive operators on Banach lattices”. It is available asarXiv:1112.1611.

Abstract. In this paper we study groups of positive operators on Banach lattices.If a certain factorization property holds for the elements of such a group, the grouphas a homomorphic image in the isometric positive operators which has the sameinvariant ideals as the original group. If the group is compact in the strong operatortopology, it equals a group of isometric positive operators conjugated by a single cen-tral lattice automorphism, provided an additional technical assumption is satisfied,for which we again have only examples. We obtain a characterization of positiverepresentations of a group with compact image in the strong operator topology, anduse this for normalized symmetric Banach sequence spaces to prove an ordered ver-sion of the decomposition theorem for unitary representations of compact groups.Applications concerning spaces of continuous functions are also considered.

4.1 Introduction and overview

In this paper we continue our efforts, initiated in Chapter 3, to develop a theoryof strongly continuous positive representations of locally compact groups in Banachlattices. In Chapter 3 we investigated positive representations of finite groups. Weshowed that a principal band irreducible positive representation of a finite groups ina Riesz space is finite dimensional, and that the representation space is necessarilyArchimedean. Furthermore, we classified such irreducible representation and showedthat each Archimedean finite dimensional positive representations is an order directsum of irreducible positive representations. Here we go a step further and considerstrongly continuous positive representations of compact groups in Banach lattices.

103

104 CHAPTER 4. COMPACT GROUPS OF POSITIVE OPERATORS

Given the relative ease with which unitary representations of compact groups canbe treated, this is the natural step to take and one would like to achieve a betterunderstanding of issues related to irreducibility and decomposition in this context.Since the image of such a representation is a group of positive operators, we examinegroups of positive operators, and since the image is compact in the strong operatortopology, we are especially interested in compact (in the strong operator topology)groups of positive operators. In fact, most of the work in this paper is aimed at abetter understanding of such compact groups. Once this is achieved, the transition tostrongly continuous positive representations with compact image is not complicated.

There are not too many papers on groups of positive operators. In [11], uniformlybounded groups of positive operators on Cc(Ω) and C0(Ω) are investigated in detail,where Ω is a locally compact Hausdorff space. These groups are studied using groupactions on the underlying space Ω and group cohomology methods. Amongst others,it is shown in [11, Example 4.1] that a strongly continuous positive representationof a compact group on C0(Ω) equals an isometric strongly continuous representationconjugated by a single central lattice automorphism, a result which we obtain as aspecial case of a more general statement, cf. Theorem 4.4.1 below.

In the case where the group G is compact in the strong operator topology, whichis the main focus of our paper, a basic result is [43, Theorem III.10.4]. It waspublished in [33], which in turn is based on unpublished lecture notes by H.P. Lotz.It gives information concerning the structure of G as well as the Banach lattice Gacts on, under the additional assumption that the action has only trivial invariantclosed ideals. Amongst others, it states that the pertinent lattice can be foundbetween C(G/H) and L1(G/H), for some closed subgroup H of G, and the groupacts as the group of left quasi-rotations induced by the natural action of G on G/H.

By studying the spectrum of lattice homomorphisms, [44] also contains someresults about groups of positive operators, in particular it is shown in [44, Corol-lary 3.10] that a uniformly bounded group of positive operators on a Banach latticeis discrete in the norm topology, a result we obtain in the special case of groupswhich are compact in the strong operator topology on certain sequence spaces, cf.Corollary 4.5.6 below.

Beyond these results not much seems to be known. Naturally, there is a theory ofone-parameter semigroups of positive operators, see, e.g., [3], but we are not aware ofissues of irreducibility or decomposition into irreducibles being considered in detailfor such semigroups.

In this paper we study groups of positive operators, or equivalently, groups oflattice automorphisms, with the property that every element can be written as aproduct of a central lattice automorphism and an isometric lattice automorphism.Remarkably enough, in the Banach lattices we consider in this paper, every latticeautomorphism is such a product. However, there are Banach lattices for which thisfails, cf. Example 4.3.1. The Banach lattices for which this holds true, as shown inthis paper, include the normalized symmetric Banach sequence spaces (Section 4.5)and spaces of continuous functions (Section 4.6). Moreover, in these spaces we havea concrete description of both the central lattice automorphisms and the isometriclattice automorphisms. In the general case, for all groups with the aforementioned

4.1. INTRODUCTION AND OVERVIEW 105

factorization property, we show that there is a group of isometric lattice automor-phisms with the same invariant ideals as the original group, cf. Theorem 4.3.2. Thisis applied to the Banach sequence spaces mentioned above, where the isometriclattice automorphisms are easily described as permutations operators, and withouttoo much effort one thus obtains a decomposition of a positive representation of anarbitrary group in such a Banach sequence space into band irreducibles, cf. The-orem 4.5.7. This result is reminiscent of the familiar decomposition theorem forstrongly continuous unitary representations of compact groups into finite dimen-sional irreducible representations, but here the representation need not be stronglycontinuous, the group need not be compact, and the (order) irreducibles can beinfinite dimensional.

Suppose the original group of automorphisms with the above factorization prop-erty is compact in the strong operator topology. As a first thought, we can equipthe Banach lattice E with an equivalent lattice norm ||| · |||, defined by

|||x||| :=∫G

‖Tx‖ dT ∀x ∈ E,

where dT denotes the Haar measure on the compact group G. Under this norm,the group G is now easily seen to be a group of isometric lattice automorphisms.However, by changing the norm, the isometries change as well, and any nice descrip-tion of the original isometries need not survive this transformation. Hence this doesnot seem useful. Instead, we impose an additional technical assumption (Assump-tion 4.3.3) on the Banach lattice. Under this assumption, which we show to holdfor normalized symmetric Banach sequence spaces with order continuous norm andspaces of continuous function, we can actually show that such a compact group isisomorphic as a topological group with the aforementioned group of isometric latticeautomorphisms with the same invariant ideals. Moreover, we can characterize suchgroups G: they are precisely the groups of the form G = mHm−1, for a uniquecompact group H of isometric lattice automorphisms, and a (non-unique) centrallattice automorphism m, cf. Theorem 4.3.8. This is especially useful whenever wehave a nice description of the central lattice automorphisms and the isometric lat-tice automorphisms, as in the spaces mentioned above. Along the same lines, onecan show that positive representations with compact image in such spaces are pre-cisely the conjugates of isometric representations, cf. Theorem 4.4.1. Moreover, inthe case that we have a positive representation in a normalized symmetric Banachsequence space with order continuous norm or in `∞ as in Theorem 4.5.7, and thepositive representation has compact image, the irreducible bands are finite dimen-sional, so that the the analogy with unitary representations of compact groups isthen complete. For positive representations with compact image in spaces of contin-uous function, one cannot in general obtain such a direct sum type decompositionas in Theorem 4.5.7, and further research is necessary to see whether there is stilla structure theorem for such representations in terms of band irreducible ones. Asa preparation, we include a number of results on the invariant closed ideals, bandsand projection bands for these representations.


The structure of this paper is as follows.

In Section 4.2 we introduce the basic notation and terminology. After establish-ing a few facts on groups of invertible operators and representations, we give a newproof of the fact that the center of a Banach lattice is isometrically algebra andlattice isomorphic to C(K), for some compact Hausdorff space K. We also obtainsome results on integrating strongly continuous center valued functions. In Sec-tion 4.3 we consider groups of lattice automorphisms for which every element is theproduct of a central lattice automorphism and an isometric lattice automorphism.We immediately obtain that there exists a group of isometric lattice automorphismshaving the same invariant ideals as the original group. Then we state the technicalAssumption 4.3.3, and under this assumption we are able to show one of our mainresults, Theorem 4.3.8, which states that every group of lattice automorphisms withthis factorization property, and which is compact in the strong operator topology,equals a group of isometric lattice automorphisms conjugated by a central latticeautomorphism. Using similar ideas, it is shown in Section 4.4 that positive rep-resentations with compact (in the strong operator topology) image are isometricpositive representations conjugated with a central lattice automorphism. We thenshow that two positive representations with compact image are order equivalent ifand only if their isometric parts are (isometrically) order equivalent. In Section 4.5we define and examine normalized symmetric Banach sequence spaces. We showthat all lattice automorphisms on such spaces are a product of a central latticeautomorphism and an isometric lattice automorphism, and that, if the space hasorder continuous norm, the technical Assumption 4.3.3 holds. Then we apply theresults from Section 4.3 and Section 4.4 to characterize compact groups of latticeautomorphisms and positive representations with compact image. We also obtainTheorem 4.5.7, the aforementioned ordered version of the decomposition theorem forunitary representations of compact groups. Finally, in Section 4.6, we examine theBanach lattice C0(Ω) for locally compact Hausdorff spaces Ω. Again we show thatall lattice automorphisms are a product of a central lattice automorphism and anisometric lattice automorphism, and that Assumption 4.3.3 holds, and we apply theresults from Section 4.3 and Section 4.4 to characterize compact groups of latticeautomorphisms and positive representations with compact image. We finish withProposition 4.6.7, which characterizes invariant closed ideals, bands and projectionbands of positive representations with compact image.

4.2 Preliminaries

In this section we discuss various facts concerning the strong operator topology,groups of invertible operators, positive representations, the center of a Banach lat-tice, and integration of strongly continuous center valued functions.

If X is a Banach space, then L(X) denotes the bounded operators on X, and thisspace equipped with the strong operator topology will be denoted by Ls(X). Subsetsof Ls(X) are always assumed to be equipped with the strong operator topology. Itfollows from the principle of uniform boundedness that compact subsets of Ls(X)


are uniformly bounded. In this topology multiplication is separately continuous, andthe multiplication is simultaneously continuous when the first variable is restrictedto uniformly bounded subsets. The next lemma is concerned with the continuity ofthe inverse.

Lemma 4.2.1. Let X be a Banach space and H ⊂ Ls(X) be a set of invertibleoperators such that H−1 is uniformly bounded. Then taking the inverse in H iscontinuous.

Proof. Let M > 0 satisfy∥∥T−1

∥∥ ≤M for all T ∈ H, and let (Ti) be a net in H thatconverges strongly to T ∈ H. Let x ∈ X, then x = Ty for some y ∈ X, and by thestrong convergence of Ti to T ,∥∥T−1

i x− T−1x∥∥ =

∥∥T−1i (Ty − Tiy)

∥∥ ≤M ‖Ty − Tiy‖ → 0.

Corollary 4.2.2. Let X be a Banach space and let H ⊂ Ls(X) be a compact setand a group of invertible operators. Then H is a compact topological group.

Proof. Compact subsets of Ls(X) are uniformly bounded, so the corollary followsfrom Lemma 4.2.1 and the simultaneous continuity of multiplication on uniformlybounded subsets of Ls(X).

As a consequence, the group H in Corollary 4.2.2 has an invariant measure, afact which will be instrumental in the proof of the key Lemma 4.3.7 below.

We continue with another lemma involving the strong operator topology, to beused in Lemma 4.3.5.

Lemma 4.2.3. Let X be a Banach space, and let A,B ⊂ Ls(X) be equipped with thestrong operator topology, such that T1S1 = T2S2 if and only if T1 = T2 and S1 = S2,for all T1, T2 ∈ A and S1, S2 ∈ B. Define pA : A · B → A and pB : A · B → B bypA(TS) := T and pB(TS) = S, for TS ∈ A ·B. Let C ⊂ A ·B be a subset such thatpA(C) is uniformly bounded and all elements of pB(C) are surjective. Then, if pBrestricted to C is continuous, pA restricted to C is continuous as well.

Proof. Let M > 0 satisfy ‖T‖ ≤M for all T ∈ A and S ∈ B with TS ∈ C. SupposepB restricted to C is continuous, and let (TiSi) be a net in C that converges stronglyto TS ∈ C, where (Ti) is a net in A and T ∈ A, and (Si) is a net in B and S ∈ B.Let x ∈ X, then x = Sy for some y ∈ X, and

‖Tix− Tx‖ = ‖TiSy − TSy‖≤ ‖TiSy − TiSiy‖+ ‖TiSiy − TSy‖≤M ‖Sy − Siy‖+ ‖TiSiy − TSy‖ ,

which converges to zero by the continuity of pB and the strong convergence of (TiSi)to TS.


Let E be a (real) Banach lattice. Being regular operators on a Banach lattice,lattice automorphisms of E are automatically bounded, and the group of latticeautomorphisms of E equipped with the strong operator topology will be denoted byAut+(E). The subgroup of isometric lattice automorphisms is denoted by IAut+(E).Equipped with the strong operator topology, we will denote these spaces by Aut+

s (E)and IAut+

s (E), and subsets of Aut+s (E) and IAut+

s (E) are always assumed to havethe strong operator topology.

Definition 4.2.4. Let G be a group and E a Banach lattice. A positive represen-tation of G in E is a group homomorphism ρ : G→ Aut+(E).

For typographical reasons, we will write ρs instead of ρ(s), for s ∈ G.Suppose ρ : G → Aut+(E) and θ : G → Aut+(F ) are positive representations in

the Banach lattices E and F . A positive operator T : E → F is called a positiveintertwiner of ρ and θ if Tρs = θsT for all s ∈ G, and ρ and θ are called orderequivalent if there exists a positive intertwiner of ρ and θ which is a lattice automor-phism. We call them isometrically order equivalent if there exists an intertwiner inIAut+(E).

We call a positive representation ρ of G in E band irreducible if the only ρ-invariant bands are 0 and E. Projection band irreducibility, closed ideal irre-ducibility, etc., are defined similarly.

In this paper we are, amongst others, concerned with subgroups of Aut+s (E).

By the above, Aut+s (E) is a group with a topology such that the multiplication is

separately continuous. Lemma 2.2.4 is a useful lemma about such groups, which westate again below.

Lemma 4.2.5. Let G and H be two groups with a topology such that right multipli-cation is continuous in both groups, or such that left multiplication is continuous inboth groups. Let φ : G→ H be a homomorphism. Then φ is continuous if and onlyif it is continuous at e.

We continue by examining the center Z(E) of a Banach lattice E, which, as in[32, Definition 3.1.1], is defined to be the set of regular operators m on E satisfying−λI ≤ m ≤ λI for some λ ≥ 0. With Zs(E) we denote Z(E) with the strongoperator topology. Central operators are often multiplication operators in concreteexamples, e.g., if 1 ≤ p ≤ ∞ and (Σ, µ) is a finite measure space, then each centraloperators m on Lp(µ) is a multiplication operator by an element of L∞(µ), cf. theexample following [32, Definition 3.1.1], and this is why we use the notation m forthese operators. The center of a Banach lattice is in all respects isomorphic toa space of continuous function, which is the context of the next proposition. Forits proof and that of Corollary 4.2.7, we recall some terminology. If E is a Banachlattice, then Orth(E) denotes the orthomorphisms of E, i.e, the order bounded bandpreserving operators ([2, Definition 2.41]). An f -algebra is a Riesz space E equippedwith a multiplication turning E into an associative algebra, such that if x, y ∈ E+,then xy ∈ E+, and if x ∧ y = 0, then xz ∧ y = zx ∧ y = 0 for all z ∈ E+ ([2,Definition 2.53]).


Proposition 4.2.6. Let E be a Banach lattice. Then the center Z(E) equipped withthe operator norm is isometrically lattice and algebra isomorphic to the space C(K)for some compact Hausdorff space K, such that the identity operator I is identifiedwith the constant function 1.

Proof. By [32, Theorem 3.1.11], the operator norm of m ∈ Z(E) is the least λ ≥ 0such that −λI ≤ m ≤ λI, i.e., it equals the order unit norm corresponding to theorder unit I, and so by [32, Proposition 1.2.13] Z(E) is an M -space with order unitI. Then the well-known Kakutani Theorem ([32, Theorem 2.1.3]) yields an isomet-ric lattice isomorphism of Z(E) with a C(K) space such that I corresponds to 1.Moreover, by [32, Theorem 3.1.12(ii)] Z(E) = Orth(E), which is an Archimedeanf -algebra by [2, Theorem 2.59]. Clearly C(K) is an Archimedean f -algebra withunit 1. By [2, Theorem 2.58] the f -algebra structure on an Archimedean f -algebrais unique, given the positive multiplicative unit, and this implies that the correspon-dence between Z(E) and C(K) must be an algebra isomorphism.

This proposition is stated in [44, Proposition 1.4], where a reference to [19] isgiven for the proof. The development of the theory since the appearance of [19]enables us to give a proof as above.

If E is a Banach lattice, then ZAut+(E) denotes the set of central lattice au-tomorphisms, i.e., ZAut+(E) = Aut+(E) ∩ Z(E). Note that ZAut+(E) does notdenote the center (in the sense of groups) of Aut+(E)! As before, ZAut+

s (E) denotesZAut+(E) equipped with the strong operator topology. In the following corollary wecollect a few properties of ZAut+(E) as they follow from the isomorphism in Propo-sition 4.2.6. If K is a compact Hausdorff space, then C(K)++ denotes the strictlypositive functions in C(K), or equivalently, the positive multiplicatively invertibleelements of C(K).

Corollary 4.2.7. Let E be a Banach lattice. Under the isomorphism Z(E) ∼= C(K)from Proposition 4.2.6, we have ZAut+(E) ∼= C(K)++. Consequently, ZAut+(E) isa group, and

Z(E) = ZAut+(E)− R+ · I = ZAut+(E)− ZAut+(E).

Proof. Suppose m corresponds to an element of C(K)++. Then m−1 correspondsto an element of C(K)++ as well, and so m is positive with a positive inverse andhence a lattice automorphism, i.e., m ∈ ZAut+(E). Conversely, let m ∈ ZAut+(E).Then m corresponds to a positive function in C(K). Since Z(E) = Orth(E), [32,Theorem 3.1.10] shows that, if m ∈ Z(E) is invertible in L(E), its inverse is in Z(E)as well. So m−1 ∈ Z(E) ∼= C(K), which is only possible if m corresponds to anelement of C(K)++. The final statement now follows from C(K) = C(K)++−R+ ·1.

The next lemma yields an isometric action of the group of lattice automorphismson the center of a Banach lattice.


Lemma 4.2.8. Let E be a Banach lattice. Conjugation by elements of Aut+(E)induces a group homomorphism from Aut+(E) into the group of isometric algebraand lattice automorphisms of Z(E). If H ⊂ Aut+

s (E) is a uniformly bounded setsuch that H−1 is also uniformly bounded and A ⊂ Zs(E) is uniformly bounded, thenthe map H × A → Zs(E) defined by (T,m) 7→ TmT−1 is continuous. Moreover,if T ∈ Aut+(E) is fixed, then m 7→ TmT−1 is a continuous algebra and latticeautomorphism of Zs(E).

Proof. Let T ∈ Aut+(E) and m ∈ Z(E), and take λ ≥ 0. Then

−λx ≤ mx ≤ λx ∀x ∈ E+ ⇔ −λT−1y ≤ mT−1y ≤ λT−1y ∀y ∈ E+

⇔ −λy ≤ TmT−1y ≤ λy ∀y ∈ E+,

hence conjugation by elements of Aut+(E) maps Z(E) isometrically into itself. Theconjugation action is obviously an algebra automorphism, and if m is positive, thenTmT−1 is positive as well, so conjugation is positive with a positive inverse, hencea lattice automorphism. The second statement follows from Lemma 4.2.1, and thecontinuity of m 7→ TmT−1 follows from the separate continuity of multiplication inthe strong operator topology.

Finally, we need a proposition for weak integration of strongly continuous centervalued functions. If X is a Banach space, (H, dh) a compact Hausdorff probabilityspace, with which we mean a compact Hausdorff space equipped with a not nec-essarily regular Borel probability measure, and g : H → X a continuous function,then [42, Theorem 3.27] shows that there exists a unique element of X, denoted by∫Hg(h) dh, defined by duality as follows:⟨∫

H

g(h) dh, x∗⟩

=

∫H

〈g(h), x∗〉 dh ∀x∗ ∈ X∗. (4.2.1)

Moreover,∫Hg(h) dh ∈ co(g(H)). By applying functionals it easily follows that

bounded operators can be pulled through the integral, and that the triangle in-equality holds.

The above vector valued integral will be used in the next proposition to definean operator valued integral. The Banach space part of the next proposition is astandard argument, which we repeat here for the convenience of the reader.

Proposition 4.2.9. Let (H, dh) be a compact Hausdorff probability space, let E bea Banach space and let f : H → Ls(E) be a continuous map. Then the operator∫Hf(h) dh : E → E, defined by(∫

H

f(h) dh

)x :=

∫H

f(h)x dh ∀x ∈ E,

where the second integral is defined by (4.2.1), defines an element of L(E) satisfying∥∥∫hf(h) dh

∥∥ ≤ suph∈H ‖f(h)‖. If S, T ∈ L(E), then

S

(∫H

f(h) dh

)T =

∫H

Sf(h)T dh. (4.2.2)

4.3. GROUPS OF POSITIVE OPERATORS 111

Moreover, if E is a Banach lattice and f(H) ⊂ Z(E), then there exist λ, µ ∈ R suchthat f(H) ⊂ [λI, µI], and for such λ and µ we have

∫Hf(h) dh ∈ [λI, µI] ⊂ Z(E).

Proof. Note that f(H) is uniformly bounded by the principle of uniform bounded-ness. The computation∥∥∥∥(∫

H

f(h) dh

)x

∥∥∥∥ =

∥∥∥∥∫H

f(h)x dh

∥∥∥∥ ≤ ∫H

‖f(h)x‖ dh ≤ suph∈H‖f(h)‖ ‖x‖

shows that the linear operator∫Hf(h) dh is bounded and that its norm satisfies

the required estimate. By applying elements of E and functionals, and using theproperties of the E-valued integral, (4.2.2) easily follows.

Now assume E is a Banach lattice and f(H) ⊂ Z(E). By the uniform bound-edness of f(H), there exist λ, µ ∈ R such that f(H) ⊂ [λI, µI]. Suppose λ andµ satisfy this relation, then we have to show that (

∫Hf(h) dh)x ∈ [λx, µx] for all

x ∈ E+, which is equivalent with∫H

λx dh ≤∫H

f(h)x dh ≤∫H

µx dh.

Now f(h)x−λx ∈ E+ for all h ∈ H by assumption, and since E+ is a closed convexset, the properties of the E-valued integral imply that

∫H

[f(h)x − λx] dh ∈ E+ aswell. The second inequality follows similarly.

4.3 Groups of positive operators

In this section we will relate certain groups of lattice automorphisms to groups ofisometric lattice automorphisms. The main assumption on these groups is that everyelement in the group can be written as a product of a central lattice automorphismand an isometric lattice automorphism. Examples of Banach lattices where this as-sumption is always satisfied are normalized symmetric Banach sequence spaces, suchas c0 and `p for 1 ≤ p ≤ ∞, where the fact that lattice automorphisms map atomsto atoms easily implies the above property, cf. Section 4.5, and spaces of continuousfunctions, where there is a well-known characterization of lattice homomorphisms interms of a multiplication operator and an operator arising from a homeomorphismof the underlying space, cf. Section 4.6. When this assumption is satisfied, we areable to show that there exists a group of isometric lattice automorphisms which hasthe same invariant ideals as the original group, cf. Theorem 4.3.2. The main result,Theorem 4.3.8, shows that, under the technical Assumption 4.3.3, for every com-pact group G of lattice automorphisms in which every element can be written as aproduct of a central lattice automorphism and an isometric lattice automorphisms,there exist a unique compact group H of isometric lattice automorphisms and anon-unique central lattice automorphism m such that G = mHm−1.

We start by showing that a certain set of lattice automorphisms is actually agroup, and, in fact, a non-trivial semidirect product. Recall that the group ofisometric lattice automorphisms of a Banach lattice E is denoted by IAut+(E), that


the group of central lattice automorphisms of E is denoted by ZAut+(E), and thatequipped with the strong operator topology these spaces are denoted by IAut+

s (E)and ZAut+

s (E). The space ZAut+(E) · IAut+(E) equipped with the strong operatortopology is denoted by ZAut+

s (E) · IAut+s (E).

Obviously, IAut+(E) is a group, and, although not quite so obvious, ZAut+(E)is also a group by Corollary 4.2.7. Now suppose mφ ∈ ZAut+(E) · IAut+(E),then (mφ)−1 = φ−1m−1 = (φ−1m−1φ)φ−1, which is in ZAut+(E) · IAut+(E) byLemma 4.2.8. In a similar vein, if m1φ1,m2φ2 ∈ ZAut+(E) · IAut+(E), thenm1φ1m2φ2 = m1(φ1m2φ

−11 )φ1φ2 ∈ ZAut+(E) · IAut+(E). Therefore the space

ZAut+(E) · IAut+(E) is a subgroup of Aut+(E).

Moreover, the representation of an element mφ ∈ ZAut+(E)·IAut+(E) is unique,and to show this it is sufficient to show that ZAut+(E)∩IAut+(E) = I. So supposem ∈ ZAut+(E) is an isometry. Then ‖m‖ =

∥∥m−1∥∥ = 1, and taking into account

the isometric isomorphism of Z(E) with a C(K) space of Proposition 4.2.6, thecontinuous function corresponding to m must be unimodular. Since this function isalso positive, it must be identically one, and so m = I.

By Lemma 4.2.8 the group IAut+(E) acts on ZAut+(E) by conjugation, andfor φ ∈ IAut+(E) and m ∈ ZAut+(E), this action will be denoted by φ(m), soφ(m) = φmφ−1. We can form the semidirect product ZAut+(E) o IAut+(E), withgroup operation

(m1, φ1)(m2, φ2) := (m1φ1(m2), φ1φ2).

Using that φ(m) is the conjugation action of φ ∈ IAut+(E) on m ∈ ZAut+(E),it easily follows that the map χ : ZAut+(E) o IAut+(E) → ZAut+(E) · IAut+(E)defined by χ(m,φ) := mφ is a group isomorphism.

All in all, it is now clear that ZAut+(E) · IAut+(E) is a subgroup of Aut+(E),and that it is isomorphic with ZAut+(E) o IAut+(E). If necessary we identifyZAut+(E) o IAut+(E) and ZAut+(E) · IAut+(E) through the map χ. The mapp : ZAut+(E) · IAut+(E)→ IAut+(E) defined by p(mφ) := φ is the projection ontothe second factor of the semidirect product, which is a group homomorphism.

In the rest of this section we will assume that the group of lattice automorphismsunder consideration is contained in ZAut+(E) · IAut+(E). For certain sequencespaces and spaces of continuous function, we will show that ZAut+(E) · IAut+(E)equals the whole group of lattice automorphisms, cf. Section 4.5 and Section 4.6,but the next example, which was communicated to us by A.W. Wickstead, showsthat there is a simple Banach lattice, not every lattice automorphism of which is aproduct of a central lattice automorphism and an isometric lattice automorphism.

Example 4.3.1. Consider R2 with the usual ordering, and with the norm definedby ‖(x, y)‖ := max|y|, |x| + |y|/2, so that it becomes a Banach lattice with thestandard unit vectors having norm one. Hence (x, y) 7→ (y, x) is the only possi-ble nontrivial isometric lattice automorphism, but this map is not isometric since‖(1, 2)‖ = 2 whereas ‖(2, 1)‖ = 5/2. Therefore (x, y) 7→ (y, x) cannot be a productof a central lattice automorphism and an isometric lattice automorphism, since it isnot a central lattice automorphism.


Theorem 4.3.2. Let E be a Banach lattice and G ⊂ ZAut+(E) · IAut+(E) a group.Then p(G) is a group of isometric lattice automorphisms, with the same invariantideals as G.

Proof. Let m ∈ Z(E) and x ∈ E+. Then there exists a positive λ such that−λx ≤ mx ≤ λx, and so mx is contained in the ideal generated by x. This factextends to all x ∈ E by writing x = x+ − x−, and so m leaves all ideals in Einvariant.

Now let mφ ∈ G, with m ∈ ZAut+(E) and φ ∈ IAut+(E), let x ∈ E and letI ⊂ E be an ideal. Since m,m−1 ∈ Z(E), by the above x ∈ I if and only if mx ∈ I,and so I is invariant for mφ if and only if I is invariant for φ = p(mφ).

We will now examine groups G ⊂ ZAut+(E)·IAut+(E) which are compact (in thestrong operator topology). In this case we can say much more than Theorem 4.3.2,if the following assumption on the Banach lattice is satisfied.

Assumption 4.3.3. If p : ZAut+s (E) · IAut+

s (E) → IAut+s (E) denotes the group

homomorphism mφ 7→ φ, then for any compact subgroup G ⊂ ZAut+s (E)·IAut+

s (E),the map p|G is continuous.

The next proposition allows us to associate compact subgroups of IAut+s (E) with

compact subgroups of ZAut+s (E) · IAut+

s (E).

Proposition 4.3.4. Let E be a Banach lattice satisfying Assumption 4.3.3. Let A bethe set of compact subgroups G ⊂ ZAut+

s (E) ·IAut+s (E), and let B be the set of pairs

(H, q), with H ⊂ IAut+s (E) a compact subgroup and q : H → ZAut+

s (E) ·IAut+s (E) a

continuous homomorphism such that p q = idH . Define α : A→ B and β : B → Aby

α(G) := (p(G), (p|G)−1), β(H, q) := q(H).

Then for each G ∈ A, G 7→ p(G) is an isomorphism of compact groups, and α andβ are inverses of each other.

Proof. If G ∈ A, then by Assumption 4.3.3 ker(p|G) is a compact subgroup ofZAut+

s (E), which is isometrically isomorphic with the group C(K)++ of strictly pos-itive continuous functions on some compact Hausdorff space K by Corollary 4.2.7.By the principle of uniform boundedness, compact subgroups of ZAut+

s (E) are uni-formly bounded, and obviously the only uniformly bounded subgroup of C(K)++ istrivial, hence p|G is a group isomorphism. Moreover it is a continuous bijection be-tween a compact space and a Hausdorff space, hence (p|G)−1 is continuous. Clearlyp (p|G)−1 = idp(G), so α is well defined.

Let G ∈ A, then β(α(G)) = β(p(G), (p|G)−1) = G. Conversely, let (H, q) ∈ B,then α(β(H, q)) = α(q(H)) = (p(q(H)), (p|q(H))

−1), and since p q = idH it followsthat p(q(H)) = H and that (p|q(H))

−1 = (p|q(H))−1 p q = q.

By the above proposition the compact subgroups G ⊂ ZAut+s (E) · IAut+

s (E)are parametrized by the pairs (H, q) of compact subgroups H ⊂ IAut+

s (E) andcontinuous homomorphism q : H → ZAut+

s (E) · IAut+s (E) satisfying pq = idH . We


will now investigate such maps q, for a given compact subgroup H of IAut+s (E). The

condition p q = idH is equivalent with the existence of a map f : H → ZAut+(E)such that q(φ) = f(φ)φ, for φ ∈ IAut+(E). We now describe the relation betweenthe continuity of f and the continuity of q.

Lemma 4.3.5. Let E be a Banach lattice satisfying Assumption 4.3.3. SupposeH ⊂ IAut+

s (E) is a compact group and q : H → ZAut+s (E) · IAut+

s (E) is a grouphomomorphism of the form q(φ) := f(φ)φ, for some map f : H → ZAut+

s (E). Thenq is continuous if and only if f is continuous.

Proof. Suppose f is continuous. Then q is the composition of f and the identity mapwith the multiplication in Aut+

s (E). Since f(H) is compact it is uniformly bounded,and multiplication in the strong operator topology is simultaneously continuous ifthe first factor is restricted to uniformly bounded sets. Therefore q is continuous.

Conversely, suppose that q is continuous. Then q(H) is a group and it is compact.Moreover q(H) is uniformly bounded, and so f(H), the set of first coordinates ofq(H), is also uniformly bounded, since ‖f(φ)‖ = ‖f(φ)φ‖ = ‖q(φ)‖ for φ ∈ IAut+(E)by the fact that φ is an isometric automorphism. Since the projection onto thesecond coordinate is continuous on q(H) by Assumption 4.3.3, Lemma 4.2.3 yieldsthe continuity on q(H) of the projection onto the first coordinate. It follows thatf is continuous as a composition of q and the projection of q(H) onto the firstcoordinate.

We continue describing the structure of maps q as above. For φ, ψ ∈ H we haveq(φψ) = f(φψ)φψ and

q(φ)q(ψ) = f(φ)φf(ψ)ψ = f(φ)φ(f(ψ))φψ.

Hence q being a homomorphism is equivalent with f(φψ) = f(φ)φ(f(ψ)) for allφ, ψ ∈ H, and such maps are called crossed homomorphisms. We will first showthat the image of such crossed homomorphisms is bounded from below.

Lemma 4.3.6. Let E be a Banach lattice, let H ⊂ IAut+s (E) be a compact group and

let f : H → ZAut+s (E) be a continuous crossed homomorphism, i.e., a continuous

map such that f(φψ) = f(φ)φ(f(ψ)) for all φ, ψ ∈ H. Then there exists an ε > 0such that f(φ) ≥ εI for all φ ∈ H.

Proof. Since f(H) is compact and hence uniformly bounded, there exists some λ > 0such that, for all φ ∈ IAut+(E),∥∥φ(f(φ−1))

∥∥ =∥∥f(φ−1)

∥∥ ≤ λ, (4.3.1)

since φ acts isometrically on Z(E) by Lemma 4.2.8. We identify ZAut+(E) withC(K)++ for some compact Hausdorff space K, using Corollary 4.2.7. Then (4.3.1)implies

0 < φ(f(φ−1)) ≤ λ (4.3.2)

pointwise on K.


By taking φ = ψ = I in the definition of a crossed homomorphism, we obtainf(I) = f(I)f(I) and so 1 = f(I). For arbitrary φ ∈ IAut+(E) we obtain

1 = f(I) = f(φφ−1) = f(φ) · φ(f(φ−1)),

and so f(φ) ≥ 1/λ pointwise on K by (4.3.2), which establishes the lemma withε = 1/λ.

To characterize the continuous crossed homomorphisms, we will use the followinglemma. It can be viewed as an analytic version of Lemma 3.4.2, which is a standardargument in group cohomology.

Lemma 4.3.7. Let E be a Banach lattice, and let H ⊂ IAut+s (E) be a compact

group. Let f : H → ZAut+s (E) be a strongly continuous map. Then f is a continuous

crossed homomorphism, i.e., a continuous map such that f(φψ) = f(φ)φ(f(ψ)),where φ(m) denotes the conjugation action of φ ∈ Aut+(E) on m ∈ ZAut+(E), ifand only if there exists an m ∈ ZAut+(E) such that f(φ) = mφ(m)−1 for all φ ∈ H.

Proof. Suppose f is a continuous crossed homomorphism. The group H is a compacttopological group by Corollary 4.2.2, and so we can equip H with its normalized Haarmeasure dψ. By Proposition 4.2.9 there exist λ, µ ∈ R such that f(H) ⊂ [λI, µI]and by Lemma 4.3.6 we may assume that λ > 0. We use Proposition 4.2.9 todefine m :=

∫Hf(ψ) dψ and also to conclude that this integral is in [λI, µI]. By

Corollary 4.2.7, [λI, µI] ⊂ ZAut+(E), and so m ∈ ZAut+(E). Then, for φ ∈ H, bythe left invariance of dψ and the fact that bounded operators can be pulled throughthe integral by (4.2.2),

φ(m) = φ

(∫H

f(ψ) dψ

)=

∫H

φ(f(ψ)) dψ

=

∫H

f(φ)−1f(φψ) dψ

= f(φ)−1

∫H

f(ψ) dψ

= f(φ)−1m,

showing that f(φ) = mφ(m)−1. Conversely, any f defined as above is continuousby Lemma 4.2.8, and such an f is easily seen to be a crossed homomorphism.

Putting everything together yields the following.

Theorem 4.3.8. Let E be a Banach lattice satisfying Assumption 4.3.3, and letG ⊂ ZAut+

s (E) · IAut+s (E) be a compact group. Then there exist a unique compact

group H ⊂ IAut+s (E) and an m ∈ ZAut+(E) such that

G = mHm−1.


Conversely, if H ⊂ IAut+s (E) is a compact subgroup and m ∈ ZAut+(E), then

G ⊂ ZAut+s (E) · IAut+

s (E) defined by the above equation is a compact subgroup ofZAut+

s (E) · IAut+s (E).

Proof. By Proposition 4.3.4, the compact subgroups of ZAut+s (E) · IAut+

s (E) areprecisely the groups q(H), where H is a compact subgroup of IAut+

s (E) and

q : H → ZAut+s (E) · IAut+

s (E)

is a continuous homomorphism satisfying p q = idH . Using Lemma 4.3.5 andLemma 4.3.7, we see that the compact subgroups of ZAut+

s (E) · IAut+s (E) are pre-

cisely the groups of the form mφ(m)−1φ : φ ∈ H = mHm−1, where H is acompact subgroup of IAut+

s (E), and m ∈ ZAut+(E). This establishes the theo-rem except for the uniqueness of H. As to this, if G = mφ(m)−1φ : φ ∈ H,for a compact group H ⊂ IAut+

s (E) and m ∈ ZAut+(E), then, in the notationof Proposition 4.3.4, G = β(H, q), where q(φ) = mφ(m)−1φ for φ ∈ H. Since(H, q) = α(β(H, q)) = α(G), this implies that H = p(G). Hence H is unique.

Note that, given a compact subgroup G ⊂ ZAut+s (E) · IAut+

s (E), the compactsubgroupH ⊂ IAut+

s (E) is unique, but the elementm ∈ ZAut+(E) in Theorem 4.3.8is obviously not unique, e.g., both m and λm for λ > 0 generate the same G.

4.4 Positive representations with compact image

In this section we will apply the results from the previous section, in particularProposition 4.3.4 and Lemma 4.3.7, to representations of groups with compact (inthe strong operator topology) image in Banach lattices satisfying Assumption 4.3.3.

Theorem 4.4.1. Let E be a Banach lattice satisfying Assumption 4.3.3, let G be agroup and let ρ : G→ ZAut+

s (E) · IAut+s (E) a positive representation with compact

image. Then there exist a unique positive representation π : G→ IAut+s (E) and an

m ∈ ZAut+(E) such that

ρs = mπsm−1 ∀s ∈ G.

The image of π is compact. Conversely, a positive representation π : G→ IAut+s (E)

with compact image and an m ∈ ZAut+(E) define a positive representation ρ withcompact image in ZAut+

s (E) · IAut+s (E) by the above equation.

In this correspondence between ρ and π, ρ is strongly continuous if and only if πis strongly continuous.

Proof. Since ρ(G) is compact, Proposition 4.3.4 applies, and so, combining this withLemma 4.3.7, p : ρ(G)→ p ρ(G) has an inverse of the form q(φ) = mφ(m)−1φ forsome m ∈ ZAut+(E) and all φ ∈ p ρ(G). We define π := p ρ, then π has compactimage, and for s ∈ G,

ρs = (q p)(ρs) = q(πs) = mπs(m)−1πs = mπsm−1.

4.5. REPRESENTATIONS IN BANACH SEQUENCE SPACES 117

This shows the existence of π. The uniqueness of π follows from the uniqueness ofthe factors in ZAut+(E) and IAut+(E) in

ρs = mπsm−1 = [mπs(m)−1]πs.

The remaining statements are now clear.

Note that, as in Theorem 4.3.8, π is unique, but m is not. Given the positiverepresentation with compact image π, m1 and m2 induce the same positive repre-sentation with compact image if and only if m−1

1 m2 commutes with πs for all s ∈ G,i.e., if and only if m−1

1 m2 intertwines π with itself.Any representation as in Theorem 4.4.1 is, by that same theorem, obviously

order equivalent to an isometric representation. In fact, we can say more. The nextproposition has the same proof as Proposition 3.4.6.

Proposition 4.4.2. Let E be a Banach lattice satisfying Assumption 4.3.3, let Gbe a group and, using Theorem 4.4.1, let ρ1 = m1π

1m−11 and ρ2 = m2π

2m−12 be

positive representations of G with compact image in ZAut+s (E) · IAut+

s (E), whereπ1 and π2 are isometric positive representations with compact image in IAut+

s (E),and m1,m2 ∈ ZAut+(E). Then ρ1 and ρ2 are order equivalent if and only if π1 andπ2 are isometrically order equivalent.

Proof. In this proof we use semidirect product notation. Suppose that ρ1 and ρ2 areorder equivalent and let T = (m,φ) ∈ Aut+(E) be a positive intertwiner. Then, forall s ∈ G,

ρ1sT = (m1π

1s(m1)−1, π1

s)(m,φ) = (m1π1s(m1)−1π1

s(m), π1sφ) (4.4.1)

Tρ2s = (m,φ)(m2π

2s(m2)−1, π2

s) = (mφ(m2)φ(π2s(m2)−1), φπ2

s), (4.4.2)

and since these are equal, φ is a positive isometric intertwiner between π1 and π2.Conversely, let φ be a positive isometric intertwiner between π1 and π2. Then,

by taking m = m1φ(m2)−1 and T = (m,φ) ∈ Aut+(E), it is easily verified that, forall s ∈ G,

(m1π1s(m1)−1π1

s(m), π1sφ) = (mφ(m2)φ(π2

s(m2)−1), φπ2s),

and so, by (4.4.1) and (4.4.2), T intertwines ρ1 and ρ2.

4.5 Positive representations in Banach sequencespaces

In this section we consider positive representations of groups in certain sequencespaces. First we show that every lattice automorphism can be written as a productof a central lattice automorphism and an isometric lattice automorphism. We arealso able to show that a large class of sequence spaces satisfy Assumption 4.3.3, and


an application of the results from Section 4.3 and Section 4.4 then yields a descrip-tion of compact groups of lattice automorphisms and of positive representations inthese spaces with compact image. Using Theorem 4.3.2, we obtain a decompositionof positive representations into band irreducibles, cf. Theorem 4.5.7. If the repre-sentation has compact image, then the irreducible bands in the decomposition inTheorem 4.5.7 are finite dimensional.

We consider normalized symmetric Banach sequence spaces E, by which we meanBanach lattices of sequences equipped with the pointwise ordering and lattice op-erations such that if x ∈ E and y is a sequence such that |y| ≤ |x|, then y ∈ Eand ‖y‖ ≤ ‖x‖, permutations of sequences in E remain in E with the same norm,and the standard unit vectors enn∈N have norm 1. Important examples are theclassical sequence spaces c0 and `p for 1 ≤ p ≤ ∞.

If x is a sequence, then x>N denotes the sequence x but with the first N coordi-nates equal to 0; similarly, x≤N denotes the sequence x with the coordinates greaterthan N equal to 0.

We will now show that a normalized symmetric Banach sequence space E sat-isfies Aut+(E) = ZAut+(E) · IAut+(E). A lattice automorphism must obviouslymap positive atoms to positive atoms, so for each T ∈ Aut+(E) and n ∈ N, thereexist a unique m ∈ N and λmn > 0 such that Ten = λmnem. Since each x ∈ E+ isthe supremum of the x≤N for N ∈ N, the linear span of atoms is order dense, andhence the above relation determines T uniquely. Therefore T can be written as theproduct of an invertible positive multiplication operator and a permutation opera-tor. We identify the group of permutation operators with S(N), so each φ ∈ S(N)corresponds to the operator defined by (φx)n := xφ−1(n) for x ∈ E and n ∈ N.The multiplication operators are identified with `∞, and by (`∞)++ we denote theset of elements m ∈ `∞ for which there exists a δ > 0 such that mn ≥ δ forall n ∈ N. We conclude that there exist an m ∈ (`∞)++ and a φ ∈ S(N) suchthat T = mφ. Conversely, an operator defined in this way is a lattice automor-phism. Obviously T is a central lattice automorphism if and only if its permuta-tion part is trivial, and so the central lattice automorphisms equal (`∞)++. ByCorollary 4.2.7 the center Z(E) equals `∞. Obviously S(N) = IAut+(E), and soAut+(E) = (`∞)++ · S(N) = ZAut+(E) · IAut+(E).

For φ ∈ S(N) and m ∈ `∞, define φ(m) ∈ `∞ by φ(m)i := mφ−1(i), the sequencem permuted according to φ. Then, for n ∈ N,

φmφ−1en = φmeφ−1(n) = φmφ−1(n)eφ−1(n) = mφ−1(n)en = φ(m)nen = φ(m)en,

which shows that m 7→ φ(m) equals the conjugation action of φ on m.We will now show that Assumption 4.3.3 holds, if E has order continuous norm.

We will actually show more, namely that p : Aut+(E) → S(N) is continuous. Thefollowing lemma is a preparation.

Lemma 4.5.1. Suppose E is a normalized symmetric Banach sequence space. Thenthe strong operator topology on S(N) ⊂ Aut+

s (E) is stronger than the topology ofpointwise convergence. If E has order continuous norm, then the topologies areequal.

4.5. REPRESENTATIONS IN BANACH SEQUENCE SPACES 119

Proof. If φ, ψ, ψ′ ∈ S(N) and (φi) is a net in S(N) converging pointwise to φ, thenψφiψ

′ → ψφψ′ pointwise, so multiplication is separately continuous in the topologyof pointwise convergence. We show that the identity map from S(N) equippedwith the strong operator topology to S(N) equipped with the topology of pointwiseconvergence is continuous, and by Lemma 4.2.5 we only have to verify continuityat the identity. Let (φi) be a net in S(N) converging strongly to the identity, andsuppose there is an n ∈ N such that φi(n) does not converge to n. If i is such thatφi(n) 6= n, then |φien − en| ≥ en and so

‖φien − en‖ ≥ ‖en‖ = 1.

It follows that φien does not converge to en, which is a contradiction. This showsthat φi converges pointwise to the identity.

Now suppose that E has order continuous norm. We will show that the iden-tity map from S(N) equipped with the topology of pointwise convergence to S(N)equipped with the strong operator topology is continuous, for which we again onlyhave to verify continuity at the identity. Let (φi) be a net in S(N) converging point-wise to the identity. Let x ∈ E and ε > 0. Since |x|>N ↓ 0 for N →∞, by the ordercontinuity of the norm we can choose N such that ‖x>N‖ = ‖|x|>N‖ < ε/2. Choosej such that φi is the identity on all indices n ≤ N , for all i ≥ j. Then, for all i ≥ j,

‖φix− x‖ ≤ ‖φi(x>N )‖+ ‖x>N‖ <ε

2+ε

2= ε,

hence φi converges strongly to the identity.

This lemma can be used to show that Assumption 4.3.3 holds, if E has ordercontinuous norm.

Lemma 4.5.2. Let E be a normalized symmetric Banach sequence space with ordercontinuous norm. Then the homomorphism p : Aut+

s (E)→ S(N) is continuous.

Proof. Again by Lemma 4.2.5 it suffices to show continuity at the identity. So let(miφi) be a net in Aut+

s (E) that converges strongly to the identity, and suppose φidoes not converge to the identity. Then by Lemma 4.5.1 there is an n ∈ N such thatφi(n) does not converge to n, and, for i such that φi(n) 6= n, we obtain

‖miφien − en‖ =∥∥∥mieφ−1

i (n) − en∥∥∥ ≥ ‖en‖ = 1.

This contradicts the assumption that miφi converges strongly to the identity, andso φi does converge to the identity.

Note that `∞ is a normalized Banach sequence space, and therefore it satisfiesAut+(`∞) = ZAut+(E) · IAut+(E). It does not have order continuous norm, butit is isometrically lattice isomorphic to C(K) for some compact Hausdorff space Kby Kakutani’s Theorem, and in Section 4.6 we will show, in Lemma 4.6.3, that suchspaces also satisfy Assumption 4.3.3.


Now we can apply the theory of the previous sections, in particular Theo-rem 4.3.8, Theorem 4.4.1 and Proposition 4.4.2, to obtain the following charac-terization of compact subgroups of Aut+

s (E) and of positive representations withcompact image.

Theorem 4.5.3. Let E be a normalized symmetric Banach sequence space withorder continuous norm or `∞, and let G ⊂ Aut+

s (E) be a compact group. Thenthere exist a unique compact group H ⊂ S(N) and an m ∈ (`∞)++ such that

G = mHm−1.

Conversely, if H ⊂ S(N) is a compact group and m ∈ (`∞)++, then G ⊂ Aut+(E)defined by the above equation is a compact subgroup of Aut+

s (E).

Theorem 4.5.4. Let E be a normalized symmetric Banach sequence space withorder continuous norm or `∞, let G be a group and let ρ : G → Aut+

s (E) be apositive representation with compact image. Then there exist a unique isometricpositive representation π : G→ S(N) and an m ∈ (`∞)++ such that

ρs = mπsm−1, ∀s ∈ G.

The image of π is compact. Conversely, any positive representation π : G → S(N)with compact image and m ∈ (`∞)++ define a positive representation ρ with compactimage by the above equation. In this correspondence between ρ and π, ρ is stronglycontinuous if and only if π is strongly continuous.

Moreover, if ρ1 = m1π1m−1

1 and ρ2 = m2π2m−1

2 are two positive representationswith compact image, where π1 and π2 are isometric positive representations withcompact image and m1,m2 ∈ (`∞)++, then ρ1 and ρ2 are order equivalent if andonly if π1 and π2 are isometrically order equivalent.

Corollary 4.5.5. Let E be a normalized symmetric Banach sequence space withorder continuous norm or `∞, and let G be a connected compact group and letρ : G→ Aut+

s (E) be a strongly continuous positive representation. Then ρs = I forall s ∈ G.

Proof. We know from Theorem 4.5.4 that ρ = mπm−1 for some strongly continuousisometric positive representation π : G → S(N) and some m ∈ (`∞)++. For eachn ∈ N, by strong continuity the orbits πsen : s ∈ G ⊂ em : m ∈ N are connected.Since for n 6= k, ‖en − ek‖ ≥ 1, the set en : n ∈ N is discrete. Hence the orbitsconsist of one point and π is trivial. But then ρ = mπm−1 is trivial as well.

Corollary 4.5.6. Let E be a normalized symmetric Banach sequence space withorder continuous norm or `∞, and let G ⊂ Aut+

s (E) be a compact group. Thenthere exists a δ > 0 such that, if T, S ∈ G, T 6= S implies ‖T − S‖ ≥ δ.

Proof. If φ 6= ψ ∈ S(N), then ‖φ− ψ‖ ≥ 1, and so the corollary follows fromTheorem 4.5.3.

4.6. POSITIVE REPRESENTATIONS IN C0(Ω) 121

In [44, Corollary 3.10], by studying the spectrum of lattice homomorphisms, thiscorollary is shown with δ =

√3 sup‖T‖ : T ∈ G, for arbitrary uniformly bounded

groups of positive operators in complex Banach lattices.Now we will examine invariant structures under these strongly continuous posi-

tive representations. In a Banach lattice with order continuous norm, the collectionof bands and the collection of closed ideal coincide by [32, Corollary 2.4.4]. Allbands in Banach sequence spaces are of the form x ∈ E : xn = 0 ∀n ∈ N \A forsome A ⊂ N; this follows easily from the characterization of bands as disjoint com-plements. Clearly this collection coincides with the collection of projection bandsand the collection of principal bands. We call a series

∑∞n=1 xn in a Riesz space un-

conditionally order convergent to x if∑∞n=1 xπ(n) converges in order to x for every

permutation π of N.

Theorem 4.5.7. Let E be a normalized symmetric Banach sequence space, let Gbe a group and let ρ : G→ Aut+(E) be a positive representation. Then E splits intoband irreducibles, in the sense that there exists an α with 1 ≤ α ≤ ∞ such that theset of invariant and band irreducible bands Bn1≤n≤α (if α < ∞) or Bn1≤n<∞(if α =∞) satisfies

x =

α∑n=1

Pnx ∀x ∈ E, (4.5.1)

where Pn : E → Bn denotes the band projection, and the series is unconditionallyorder convergent, hence, in the case that E has order continuous norm, uncondi-tionally convergent.

Moreover, if ρ has compact image and E has order continuous norm or E = `∞,then every invariant and band irreducible band is finite dimensional, and so α =∞.

Proof. We define the isometric positive representation π := p ρ : G→ S(N), whichhas the same invariant bands as ρ by Theorem 4.3.2. It follows immediately fromthe above parametrization of the bands of E that the irreducible bands are given bythe orbits π(G)en of the standard unit vectors en. In the case that ρ(G) is compactand E has order continuous norm or E = `∞, the map p|ρ(G) is continuous, so theseorbits p(ρ(G))en are compact in E, and hence consist of finitely many standard unitvectors, and so the irreducible bands are finite dimensional and there are countableinfinitely many of them. The unconditional order convergence of the series (4.5.1)follows from the fact that |π(x)|≥N ↓ 0 as N →∞ for any permutation π of N.

In the order continuous case, the series (4.5.1) need not be absolutely convergent,which can be seen by taking the trivial group acting on a normalized symmetricBanach sequence space with order continuous norm not contained in `1 and takingan x ∈ E not in `1.

4.6 Positive representations in C0(Ω)

In this section we consider the space C0(Ω), where Ω is a locally compact Haus-dorff space. First we show that every lattice automorphism of C0(Ω) is the product


of a central lattice automorphism and an isometric lattice automorphism. We willalso show that C0(Ω) satisfies Assumption 4.3.3, from which we obtain a charac-terization of compact groups and representations of positive groups with compactimage, using the results from Section 4.3 and Section 4.4. As we will explain below,contrary to the sequence space case one cannot expect to find a direct sum typedecomposition of an arbitrary strongly continuous positive representation into bandirreducibles for general C0(Ω). More investigation is necessary to determine whetherthese representations are still built up, in an appropriate alternative way, from bandirreducible representations. As a preparation for such future research, we collectsome results about the structure of invariant ideals, bands and projection bands, cf.Proposition 4.6.7.

Analogously to the sequence space case from Section 4.5, we will start by show-ing that Aut+(C0(Ω)) = ZAut+(C0(Ω)) · IAut+(C0(Ω)). Elements of Homeo(Ω),the group of homeomorphisms of Ω, are viewed as elements of Aut+

s (C0(Ω)) byφx := x φ−1 for x ∈ C0(Ω). The set of multiplication operators by continu-ous bounded functions is denoted by Cb(Ω), and by Cb(Ω)++ we denote the el-ements m ∈ Cb(Ω) such that there exists a δ > 0 such that m(ω) ≥ δ for allω ∈ Ω. It follows from [32, Theorem 3.2.10] that every T ∈ Aut+(C0(Ω)) canbe written uniquely as a product of an element m ∈ Cb(Ω)++ and an operatorφ ∈ Homeo(Ω), so T = mφ. Conversely, any T defined in this way is a latticeautomorphism. It is easy to see that T ∈ Z(C0(Ω)) if and only if its part inHomeo(Ω) is trivial, so Cb(Ω)++ is the group of central lattice automorphisms, andby Corollary 4.2.7, Z(C0(Ω)) ∼= Cb(Ω). Obviously Homeo(Ω) = IAut+(E), and soAut+(C0(Ω)) = Cb(Ω)++ ·Homeo(Ω) = ZAut+(C0(Ω)) · IAut+(C0(Ω)).

For φ ∈ Homeo(Ω) and m ∈ Cb(Ω)++, define φ(m) := m φ−1 ∈ Cb(Ω)++.Then, for ω ∈ Ω and x ∈ C0(Ω),

φmφ−1x(ω) = mφ−1x(φ−1(ω)) = m(φ−1(ω))φ−1x(φ−1(ω)) = m(φ−1(ω))x(ω),

so φ(m) equals the conjugation action of φ on m.Our next goal is to show that Assumption 4.3.3 is satisfied, and for that we have

to examine Homeo(Ω). The topological structure of Homeo(Ω) can be describedby the following lemma, the proof of which is given by [51, Definition 1.31], [51,Lemma 1.33] and [51, Remark 1.32].

Lemma 4.6.1. The strong operator topology on Homeo(Ω) equals the topology withas subbasis elements of the form

U(K,K ′, V, V ′) := φ ∈ Homeo(Ω) : φ(K) ⊂ V and φ−1(K ′) ⊂ V ′

with K and K ′ compact and V and V ′ open. A net (φi) in Homeo(Ω) convergesto φ ∈ Homeo(Ω) if and only if ωi → ω ∈ Ω implies that φi(ωi) → φ(ω) andφ−1i (ωi)→ φ−1(ω).

Before we can show the validity of Assumption 4.3.3 for C0(Ω), we need a smalllemma.


Lemma 4.6.2. Let (mi) be a net in Cb(Ω)++ and (φi) be a net in Homeo(Ω) suchthat miφi converges strongly to the identity. If ωi → ω ∈ Ω, then φ−1

i (ωi)→ ω.

Proof. Suppose that there exists a net (ωi) converging to ω such that φ−1i (ωi) does

not converge to ω. By passing to a subnet we may assume that there exists an openneighborhood U of x such that φ−1

i (ωi) /∈ U for all i. Take a compact neighborhoodK of x such that K ⊂ U , then by passing to a subnet we may assume that ωi ∈ Kfor all i. By [51, Lemma 1.41], a version of Urysohn’s Lemma, there exists a functionx ∈ Cc(Ω) such that x is identically one on K and zero outside of U . Then

‖miφix− x‖ ≥ |[miφix](ωi)− x(ωi)|= |mi(ω)x(φ−1

i (ω))− x(ωi)| = |0− 1| = 1,

which contradicts the strong convergence of miφi to the identity.

Recall that p : Aut+(C0(Ω))→ Homeo(Ω) denotes the map mφ 7→ φ.

Lemma 4.6.3. Let G be a compact subgroup of Aut+s (C0(Ω)). Then the map

p|G : G→ Homeo(Ω) is continuous.

Proof. Again by Lemma 4.2.5 it is enough to show continuity at the identity. Sosuppose that (mi) is a net in Cb(Ω)++ and (φi) is a net in Homeo(Ω) such that (miφi)is a net in G converging strongly to the identity. By Lemma 4.2.1 the inverse map inG is continuous, and (miφi)

−1 = φ−1i (m−1

i )φ−1i also converges to the identity. We

have to show that φi converges to the identity in Homeo(Ω). Let ωi → ω ∈ Ω. Byapplying Lemma 4.6.2 twice we obtain φ−1

i (ωi)→ ω and φi(ωi) = (φ−1i )−1(ωi)→ ω.

This is precisely what we have to show by Lemma 4.6.1.

Hence Assumption 4.3.3 is satisfied, and once again we can apply the resultsof Section 4.3 and Section 4.4, in particular Theorem 4.3.8, Theorem 4.4.1 andProposition 4.4.2, to obtain the following.

Theorem 4.6.4. Let Ω be a locally compact Hausdorff space and suppose thatG ⊂ Aut+

s (C0(Ω)) is a compact group. Then there exist a unique compact groupH ⊂ Homeo(Ω) and an m ∈ Cb(Ω)++ such that

G = mHm−1.

Conversely, any compact group H ⊂ Homeo(Ω) and m ∈ Cb(Ω)++ define a compactgroup G ⊂ Aut+

s (C0(Ω)) by the above equation.

Theorem 4.6.5. Let Ω be a locally compact Hausdorff space, let G be a group andlet ρ : G → Aut+

s (C0(Ω)) be a positive representation with compact image. Thenthere exist a unique isometric positive representation π : G → Homeo(Ω) and anm ∈ Cb(Ω)++ such that

ρs = mπsm−1 ∀s ∈ G.

The image of π is compact. Conversely, a positive representation π : G→ Homeo(Ω)and an m ∈ Cb(Ω)++ define a positive representation ρ with compact image by the


above equation. In this correspondence between ρ and π, ρ is strongly continuous ifand only if π is strongly continuous.

Moreover, if ρ1 = m1π1m−1

1 and ρ2 = m2π2m−1

2 are two positive representationswith compact image, where π1 and π2 are isometric positive representations withcompact image and m1,m2 ∈ Cb(Ω)++, then ρ1 and ρ2 are order equivalent if andonly if π1 and π2 are isometrically order equivalent.

A part of this result is obtained in [11, Example 4.1], by using group cohomologymethods on group actions on the set Ω.

Contrary to the sequence space case, our results do not, in general, lead to a de-composition of positive representations with compact image into band irreducibles.Indeed, if the trivial group acts on C[0, 1], then every band in C[0, 1] is invariant,but since every nonzero band properly contains another nonzero band, there are noinvariant band irreducible bands. However, we can still say something about thevarious invariant structures of such representations, and for this we need a charac-terization of these structures in C0(Ω).

Lemma 4.6.6. Let Ω be a locally compact Hausdorff space. Every closed idealI ⊂ C0(Ω) is of the form

I = IS = f ∈ C0(Ω) : f(S) = 0

for a unique closed S ⊂ Ω. Hence S 7→ IS is an inclusion reversing bijection betweenthe closed subsets of Ω and the closed ideals of C0(Ω). The ideal IS is a band if andonly if S is regularly closed, i.e., S = int(S), and it is a projection band if and onlyif S is clopen.

Proof. [32, Proposition 2.1.9] and [32, Corollary 2.1.10] show this statement for Ωcompact, and the proof also works if Ω is locally compact.

The next result may serve as in ingredient in the further study of positive repre-sentations in C0(Ω). If π : G → Homeo(Ω) is a map, then a subset S ⊂ Ω is calledπ-invariant if πs(S) ⊂ S for all s ∈ G.

Proposition 4.6.7. Let G be a group and let ρ : G → Aut+(C0(Ω)) be a positiverepresentation, and define π := p ρ : G→ Homeo(Ω). Then the map S 7→ IS fromLemma 4.6.6 restricts to a bijection between the π-invariant closed subsets S ⊂ Ωand the ρ-invariant closed ideals of C0(Ω). By further restriction, this induces abijection between the π-invariant regularly closed subsets of Ω and the ρ-invariantbands of C0(Ω), and between the π-invariant clopen subsets of Ω and the ρ-invariantprojection bands of C0(Ω).

Proof. By Theorem 4.3.2 the invariant ideals of ρ(G) are the same as the invariantideals of π(G) ⊂ Homeo(Ω), and so we may assume that ρ = π. Let S ⊂ Ω be aπ-invariant closed subset. Then, for all f ∈ IS , s ∈ G and ω ∈ Ω,

πsf(ω) = f(π−1s (ω)) = 0,


and so IS is a π-invariant closed ideal of C0(Ω).Conversely, let IS be a π-invariant closed ideal of C0(Ω) for some closed S ⊂ Ω.

Let ω ∈ Ω, then for all f ∈ IS and s ∈ G, f(πs(ω)) = π−1s f(ω) = 0, and so

πs(ω) ⊂ S, hence S is π-invariant. This shows the statement about closed ide-als, and the statements about bands and projection bands follow immediately fromLemma 4.6.6.

Acknowledgements

The authors thank Anthony Wickstead for providing Example 4.3.1.

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Dankwoord

Op de eerste plaats wil ik mijn begeleider Marcel de Jeu bedanken voor zijn inzet enonze prettige samenwerking, in het bijzonder voor zijn grote hoeveelheid nauwkeurigcommentaar op mijn schrijfstijl en zijn vele nuttige en wiskundig precieze opmerkin-gen en toevoegingen. Ook wil ik graag al mijn collega’s van het MathematischInstituut bedanken, in het bijzonder mijn medepromovendus Miek Messerschmidt,met wie ik vele interessante discussies heb gevoerd, en Lenny Taelman, voor zijncommentaar over groepen van eindigdimensionale positive operatoren. Ook wil ikgraag mijn ouders bedanken voor hun steun gedurende mijn studie en promotie.

131

Samenvatting

Gemotiveerd door onder andere de quantummechanica zijn sterk continue unitairerepresentaties van groepen intensief bestudeerd. Voor zulke representaties van com-pacte groepen is er een decompositie in eindigdimensionale irreducibele represen-taties als orthogonale directe som mogelijk. Als de groep niet compact is en derepresentatieruimte een separabele Hilbertruimte is, is er nog steeds een decomposi-tie in irreducibelen in termen van directe integralen van Hilbertruimtes en directeintegralen van representaties.

Als een groep werkt op een verzameling induceert dat vaak een unitaire repre-sentatie in een ruimte van L2-functies. In dergelijke situaties krijgen we dan meestalook sterk continue representaties in Lp-ruimtes en ruimtes van continue functies.Dit zijn representaties in Banachruimtes die geen unitaire representaties zijn. Tochwillen we graag dit soort representaties begrijpen. In het bijzonder, is er voor ditsoort representaties ook een decompositie in irreducibelen mogelijk?

Dit proefschrift is een bijdrage aan de theorie van zulke representaties. Hetbestaat uit twee delen. In het eerste gedeelte bekijken we gekruiste producten, enin het tweede deel bekijken we positieve representaties. We beschouwen nu eerst heteerste gedeelte.

In de representatietheorie zijn er, gegeven een groep, vaak algebra’s beschikbaarmet de eigenschap dat representaties van de groep in bijectie zijn met representatiesvan de algebra. Een voorbeeld is de groepsalgebra van een eindige groep, waarvanrepresentaties in een vectorruimte in bijectie zijn met representaties van de groep indie vectorruimte. In de unitaire theorie is er ook zo’n algebra, de groep C∗-algebra,waarvan de zogenaamde niet-degenereerde representaties in een Hilbertruimte inbijectie zijn met unitaire representaties van de groep in die Hilbertruimte. Dezealgebra is erg nuttig gebleken in de decompositietheorie van unitaire representaties.

Dit is de reden dat we, gegeven een lokaal compacte groep, ook een algebratot onze beschikking willen hebben waarvan een klasse van zijn representaties inBanachruimtes in bijectie is met een klasse van sterk continue representaties van degroep. De reden dat we niet alle representaties beschouwen is als volgt. Er is maareen oneindigdimensionale separabele Hilbertruimte op isomorfie na, namelijk `2,terwijl er een grote diversiteit aan oneindigdimensionale separabele Banachruimtesis. De hierboven genoemde algemene decompositiestelling is dus in feite een stellingover een ruimte. Het is te lastig om alle representaties in alle Banachruimtes tebekijken. Het zou beter zijn om voor specifieke klassen van representaties specifieke

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Banachalgebra’s te hebben met de juiste bijectie-eigenschap. Met andere woorden,de constructie van de groep C∗-algebra moet gegeneraliseerd worden, zodanig datdeze aangepast kan worden aan de klasse representaties waarin men geınteresseerdis, in plaats van alleen voor unitaire representaties te werken.

Dit is in Hoofdstuk 2 gedaan, waarin we een nog algemener object dan de groepC∗-algebra generaliseren, namelijk de gekruist product C∗-algebra. Gegeven eenC∗-dynamisch systeem is de gekruist product C∗-algebra een C∗-algebra, waarvande niet-gedegenereerde representaties in bijectie zijn met de zogenaamde covarianterepresentaties van het C∗-dynamische systeem. In ons geval kijken we naar een Ba-nachalgebra dynamisch systeem, en zoals hierboven uitgelegd is, is een nieuwe vari-abele hierbij een klasse R van covariante representaties, die in het C∗-geval gelijk isaan alle covariante representaties. Het hoofdresultaat uit dit hoofdstuk laat zien dater een Banachalgebra bestaat, de gekruist product Banachalgebra, waarvan de niet-gedegenereerde algebrarepresentaties in bijectie zijn met de zogenaamde R-continuecovariante representaties van het oorspronkelijke Banachalgebra dynamische sys-teem, een in het algemeen grotere klasse dan R. In het C∗-geval vallen deze klassensamen. Als we nu een Banachalgebra dynamisch systeem nemen met een trivialeBanachalgebra, dan krijgen we een algebra waarvan de niet-gedegenereerde repre-sentaties in bijectie zijn met een klasse van representaties van een lokaal compactegroep.

Dit hebben we toegepast op specifieke situaties, waardoor we nu klassen vangroepsrepresentaties in Banachruimtes kunnen bekijken door de algebrarepresen-taties van het corresponderende Banachalgebra gekruist product te bestuderen (hetpassende analogon van de groep C∗-algebra), die in principe makkelijker te begrijpenzijn aangezien Banachalgebra’s veel meer functionaalanalytische structuur hebbendan groepen.

In Hoofdstuk 3 en Hoofdstuk 4 bestuderen we positieve representaties in Riesz-ruimtes en Banachroosters, waarbij we in het bijzonder geınteresseerd zijn in decom-posities in irreducibelen. Veel representaties uit de praktijk, bijvoorbeeld represen-taties geınduceerd door groepsacties op verzamelingen, zijn positieve representaties.In Hoofdstuk 3 bekijken we het eenvoudigste geval: eindige groepen.

De eerste vraag is wat het juiste concept van irreducibiliteit is in de geordendecontext. In de unitaire theorie is irreducibiliteit van representaties equivalent metindecomposabiliteit, waarmee bedoeld wordt dat de representatie niet orthogonaalopgesplitst kan worden in twee deelruimtes die invariant zijn onder de representatie.In onze geordende context blijkt dat orde indecomposabiliteit van representaties tezijn, d.w.z. dat de ruimte niet geordend opgesplitst kan worden in twee deelruimtesdie invariant zijn onder de representatie.

We zijn dus in het bijzonder geınteresseerd in orde indecomposabele representa-ties, en een natuurlijke vraag is of dergelijke representaties van eindige groepen altijdeindigdimensionaal zijn. In de unitaire theorie is de analoge uitspraak altijd waaren het bewijs is eenvoudig. Dit unitaire bewijs werkt echter niet in de geordendecontext. Een van de hoofdresultaten van Hoofdstuk 3 laat echter zien dat ook in degeordende context het antwoord op deze vraag altijd bevestigend is, mits de ruimte

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waarin we kijken Dedekind volledig is. Het bewijs gebruikt inductie naar de ordevan de groep en toont overigens aanzienlijk meer aan dan we hier noemen.

In dit hoofdstuk karakteriseren we ook de groep van roosterautomorfismen vanRn met de kanonieke ordening, voor n ∈ N. Alle eindigdimensionale Archimedi-sche Rieszruimtes zijn isomorf met Rn voor een n ∈ N. Deze groep blijkt eensemidirect product te zijn van de groep van diagonaalmatrices met strikt positievediagonaalelementen en de groep van permutatiematrices. Hieruit volgt uiteindelijkdat alle eindige groepen van roosterautomorfismen groepen van permutatiematriceszijn, geconjugeerd door een diagonaalmatrix met strikt positieve diagonaalelemen-ten. Dit impliceert dat representaties van eindige groepen in eindigdimensionaleArchimedische Rieszruimtes op een soortgelijke manier beschreven kunnen worden.Uiteindelijk verkrijgen we een beschrijving van de ordeduaal van een eindige groep,d.w.z. van de equivalentieklassen van orde-equivalente orde indecomposabele posi-tieve representaties in Dedekind volledige Rieszruimtes, in termen van groepsactiesop eindige verzamelingen. We laten ook zien dat een positieve representatie vaneen eindige groep in een eindigdimensionale Archimedische Rieszruimte altijd uniekopsplitst in orde indecomposabele representaties.

Het blijkt verder dat er voorbeelden zijn van orde equivalente orde indecompo-sabele representaties met hetzelfde karakter, dus karakters bepalen geen positieverepresentaties zoals in het unitaire geval. Ook hebben we inductie in het geordendegeval bekeken, waarvan de categorietheoretische aspecten grotendeels hetzelfde zijnals in het niet geordende geval. De multipliciteitsversie van Frobenius reciprociteitblijkt echter niet te gelden.

In Hoofdstuk 4 nemen we de volgende stap: na de eindige groepen gaan weover op sterk continue positieve representaties van compacte groepen. Om dezete bestuderen bekijken we groepen van roosterautomorfismen die compact zijn inde sterke operatortopologie. We nemen aan dat deze groepen bevat zijn in hetproduct, dat weer een semidirect product is, van de groep van centrale roosterauto-morfismen en de groep van isometrische roosterautomorfismen. Dit is gemotiveerddoor het hierboven beschreven semidirect product in het eindigdimensionale geval.Voor genormaliseerde symmetrische Banach rijtjesruimtes met orde continue normen voor ruimtes van continue functies geldt zelfs dat de hele groep van roosterauto-morfismen gelijk is aan dit semidirecte product, en dus geldt bovenstaande aannamevoor compacte groepen van roosterautomorfismen altijd.

Onder een milde technische aanname die geldt in bovenstaande rijtjesruimtes enruimtes van continue functies blijkt weer dat zulke compacte groepen van roosterau-tomorfismen gelijk zijn aan groepen van isometrische roosterautomorfismen, gecon-jugeerd met een centraal roosterautomorfisme. Dit levert weer een goede beschrijvingop van sterk continue positieve representaties van compacte groepen, en als we dittoepassen op rijtjesruimtes, krijgen we een resultaat analoog aan het unitaire geval inhet begin van deze samenvatting: elke sterk continue positieve representatie van eencompact groep in een genormaliseerde symmetrische Banach rijtjesruimte met ordecontinue norm of in `∞ heeft een decompositie als orde directe som van eindigdi-mensionale orde indecomposabele deelrepresentaties. Deze klasse van rijtjesruimtesbevat de klassieke rijtjesruimtes c0 en `p voor 1 ≤ p <∞.

Curriculum Vitae

Marten Wortel werd op 15 juli 1981 geboren in Delft, waar hij in 1999 zijn gymnasiumdiploma behaalde aan het Stanislascollege. In datzelfde jaar begon hij de studietechnische wiskunde in Delft. Eind 2007 studeerde hij daar af, met de masterscriptieVector-valued Function Spaces and Tensor Products onder begeleiding van prof. dr.B. de Pagter. Vervolgens begon hij in december 2007 als Assistent in Opleidingaan het Mathematisch Instituut in Leiden bij prof. dr. S.M. Verduyn Lunel onderdagelijkse begeleiding van dr. M.F.E. de Jeu. Hij bezocht conferenties in Leidenen Kopenhagen, en in Madrid, Dresden, Tianjin en Athene bezocht hij conferentieswaar hij ook voordrachten hield over zijn werk. Dit proefschrift is het resultaat vanzijn promotieonderzoek.

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group representations in banach spaces and banach...

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